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Professor of Physics in the 

University of Iowa 




£$o Fourth Avenue 


Copyright, 1932, 1933, by 

All Rights Reserved 
This book, or any parts thereof, may not 
be reproduced in any form without 
written permission from the publishers. 

Special Edition, March 1932 

Regular Edition 
Published, February 1933 
Reprinted, February 1937 



The accompanying text is an elementary treatise that under- 
takes to consider the most common phenomena in acoustics. 
The content assumes no previous preparation in physics, and 
utilizes very few mathematical expressions. The limitation in 
preparation of the student is met by the insertion in the text of 
the meaning of each technical term at the point where it is first 

The absence of mathematics places an increased responsibility 
upon language in presenting a clear analysis of all the phenomena. 
Thus at many points the writing is necessarily condensed and 
requires careful reading and re-reading. If the student will an- 
ticipate this type of effort he will have no serious difficulty. The 
text does not survey the field rapidly as most elementary texts 
do; it endeavors to study each topic with a thoroughness some- 
what unexpected in a nonmathematical text. The student will 
secure an acquaintance that will not only serve as a background 
for any professional work involving acoustics, but also as valu- 
able information that can be applied with success. As a matter 
of fact, the viewpoint of the book is utility in the broadest sense, 
including culture. The historical aspects of the subject are 
largely omitted to make room for the detailed explanations and 
analyses which are regarded as more important. 

The number of students who need such a background and 
yet who cannot afford the time for mathematical studies is rap- 
idly increasing. While these have been prominently in the 
author's mind, yet it is evident that the book can also be used 
in intermediate courses in physics. Moreover, the amount of 
acoustics in the usual elementary course in physics is so small 
that students with and without previous preparation in elemen- 
tary physics can use this text in the same class. But it is pref- 
erable that the course be offered in the junior and senior years 
rather than in the first year of college. 



Numerous demonstration experiments are suggested through- 
out. These and others which can be introduced by the instructor 
will prove invaluable. 

The preparation of this text has extended over several years 
in connection with classes consisting for the most part of students 
specializing in music, speech and psychology. The attitude they 
have shown toward acquiring a clear understanding of acoustics 
and the pleasure they have derived from the nonmathematical 
analyses of phenomena have supplied the incentive for the revi- 
sions and final preparation of the manuscript. 

I take pleasure in acknowledging my indebtedness to the 
students who have from time to time given excellent criticism 
and especially to my colleagues, Dr. P. G. Clapp, Director of 
Music, who prepared Section 14.6, and Dr. C. J. Lapp, who has 
critically examined the entire manuscript. 

George Walter Stewart 



Sound Waves 


Acoustics I 

Waves 3 

Properties of Waves 4 

A " Wave " in a Helix 7 

Different Aspects of a Wave 8 

Gas as a Medium for Sound Waves 10 

7 Representation of a Sound Wave 10 

8 Velocity 15 

9 A Variation of Velocity 18 

10 Frequency and Wave-Length 20 

1 1 Doppler's Principle 20 

12 Velocity of the "Particle" of the Medium 21 

Questions 21 


Reflection and Absorption in Auditoriums 

2. 1 Reflection at a Plane Surface 23 

2.2 Echo 25 

2.3 Reverberation 26 

2.4 Absorption 26 

2.5 Reverberation in a Room 27 

2.6 Modern Absorbing Materials 30 

2.7 Absorption Coefficients 32 

2.8 Absorption Coefficients and Frequency 32 

2.9 Other Effects in an Auditorium 34 

Questions 34 


Acoustic Reflectors 

3.1 Nature of Interference 3$ 

3.2 Huyghens' Principle 36 

3.3 A Beam of Sound 37 

3.4 Acoustic Plane Reflector 38 

3.5 Acoustic Parabolic Mirror 39 

3.6 Interference in Auditoriums 40 

3.7 Selective Property of Reflectors 42 

3.8 The Pinnae as Reflectors 46 



3.9 Acoustic Horns as Reflectors 47 

Questions 47 

Refraction and Diffraction 

4.1 Variations of Velocity in the Atmosphere 49 

4.2 Effect of Temperature 49 

4.3 Effect of the Wind 52 

4.4 Speaking in the Wind 53 

4.5 Silence Areas 55 

4.6 Refraction and Scattering of Airplane Noises 56 

4.7 Diffraction 57 

4.8 Diffraction about the Head of a Speaker 58 

4.9 Diffraction about the Head of an Auditor 61 

4.10 Change of Quality by Diffraction 63 

4. 1 1 Principle of Least Time 63 

4.12 Passage of Aerial Waves about the Earth ,. . 63 

Questions 64 


Phase Change at Reflection 

5. 1 Phase Change 6$ 

5.2 Reflection without Change of Phase 65 

5.3 Reflection with Change of Phase 66 

5.4 Interesting Cases of Reflection in Gases 68 

5.5 The Image in Reflection without Change of Phase 70 

5.6 The Image in Reflection with Change of Phase 71 

5.7 Reflection at a Change in Area of a Conduit 71 

5.8 Cause of Reflection 73 

5.9 Reflection at an Open End of a Pipe 75 

5.10 Reflection at a Closed End of a Pipe 77 

5. 1 1 Total Reflection at an Interface . . . . 77 

5.12 Absorption Along a Conduit 78 

Questions 78 


6.1 General Phenomenon of Resonance . , . . . 80 

6.2 Plane Stationary Waves 81 

6.3 Stationary Waves in a Cylindrical Pipe Closed at One 

End . . . . . 84 

6.4 Resonance 86 

6.5 Emission of Sound Increased by Resonance . . . 87 



6.6 Resonance in a Volume having an Orifice 88 

6.7 Resonance of the Voice 9° 

6.8 Resonance in Cylindrical Pipes 9° 

6.9 End Correction of an Open Pipe 9 1 

6.10 Resonance in Conical Megaphones 9 2 

6.1 1 Megaphones not Conical 94 

6.12 Stationary Waves in General 94 

6.13 Resonance in Musical Instruments 95 

6.14 Resonance in Buildings 95 

Questions 9^ 


Musical Sounds 

7. 1 Musical Tones 97 

7.2 The Vibration of a String 97 

7.3 Measurement of Relative Intensities of Fundamental and 

Overtones 99 

7.4 Instrumental Quality 101 

7.5 Sounds from Various Instruments 101 

7.6 Recognition of Phase Differences of the Components 104 

Questions • io 5 


The Nature of Vowel Sounds 

8.1 The Nature of Speech Sounds . 106 

8.2 The Vowels Used 106 

8.3 Characteristic Regions are Resonance Regions 1 1 1 

8.4 Clearness of Enunciation of Vowels 114 

8.5 Variation in Vowel Sounds 114 

Questions • • • • IJ 6 


Certain Physical Factors in Speech 

9.1 Energy Distribution 117 

9.2 Useful Energy Distribution 119 

9.3 Speech Energy • • 1 2 l 

Questions •• • 1*2 



10. 1 Energy Required for Minimum Audibility 123 

10.2 Limits of Audibility 1 2 4 



10.3 Deafness Defined in Dynamical Units I2 r 

10.4 Types of Deafness I2 * 

10.5 Loudness I27 

10.6 Weber's Law — Fechner's Law — Sensation Units 128 

10.7 Minimum Perceptible Difference in Intensity 130 

10.8 Audibility of a Tone Affected by a Second Tone: Masking 

Effe ct. I3 o 

10.9 Hearing in the Presence of Noise j o 2 

10.10 Minimum Time for Tone Perception 1^3 

10.11 Minimum Perceptible Difference in Frequency 133 

10. 1 2 The Vibrato : *. 

10.13 Loudness of Complex Sounds jor 

10.14 Combinational Tones jor 

10.15 Frequencies Introduced by Asymmetry 13 r 

10.16 The Ear an Asymmetrical Vibrator 136 

10.17 Use of Combinational Tones in the Organ 137 

10. 1 8 Pressure of Sound Waves 138 

10.19 Intermittent Tones 138 

10.20 Intensity and Pitch of a Blend of Sounds , 139 

Questions j^q 


Binaural Effects 

1 Binaural Intensity Effect 140 

2 Binaural Phase Effect 142 

3 Phase Effect with Complex Tones 146 

4 Utilization of the Binaural Phase Effect 146 

5 Complexity of Factors in Actual Localization 146 

6 Demonstration of Binaural Phase Effect 147 

7 Binaural Beats 148 

Questions 148 


Acoustic Transmission 

1 2. 1 Transmission of Energy from One Medium to Another . . . 149 

12.2 Architectural Acoustics 151 

12.3 Machinery Noises 151 

1 2.4 Case of Three Media 152 

12.5 Constrictions and Expansion in Conduits 153 

12.6 The Stethoscope 154 

12.7 Non-reflecting Conduit Junctions 155 

1 2.8 Velocity of Sound in Pipes 155 

12.9 Decay of Intensity in Pipes 156 

Questions 158 



Selective Transmission 


13. i Interference Tube of Herschel and Quincke 159 

13.2 Theory of a Closed Tube as a Side Branch 164 

13.3 Helmholtz Resonator as a Side Branch 165 

13.4 Action of an Orifice 166 

13.5 Acoustic Wave Filters 168 

13.6 Moving Nodes 17° 

Questions 17 1 

Musical Scales 

14.1 The Diatonic Scale 17 2 

14.2 Mean Temperament 173 

14.3 Frequency 174 

14.4 Nomenclature 175 

14.5 Musical Intervals 176 

14.6 Production of Music in the Natural Scale 176 

Questions 176 

Musical Instruments, the Voice and Other Sound Sources 

15. 1 Development of Musical Instruments 177 

15.2 Production of Sound, General 177 

15.3 Production of Sound by Strings 179 

1 5.4 Production of Sound by Reeds 1 80 

15.5 Production of Sound by Air Blasts 181 

15.6 Harmonics and Overtones 182 

15.7 Peculiarity of Action of Several Instruments 183 

15.8 Emission of Sound from the Clarinet 185 

15.9 Production of the Voice 185 

1 5. 10 Frequency of Pipes 186 

15. 11 Aeolian Harp 186 

15.12 Singing Flames 187 

15.13 Singing Tubes 187 

15.14 Sensitive Flames and Jets 188 

15.15 Tones from Membranes 188 

15.16 Sound Waves in a Solid 190 

15.17 Vibration of Bells 191 

15.18 Carillons and Chimes 192 

15.19 Acoustic Power Output 193 

15.20 Modern Loud Speakers 193 

Questions 1 94 

To the Student who uses this Texiboo\: 

This textbook represents many years of 
learning and experience on the part of the 
author. It does not treat of an ephemeral 
subject, but one which, since you are study- 
ing it in college, you must feel will have a 
use to you in your future life. 

Unquestionably you will many times in 
later life wish to refer to specific details 
and facts about the subject which this book 
covers and which you may forget. How 
better could you find this information than 
in the textbook which you have studied from 
cover to cover? 

Retain it for your reference library. You 
will use it many times in the future. 

The Publishers. 


i.i. Acoustics. — Interest in acoustics has been increasing 
very rapidly during the past two decades. This has not occurred 
because of advances made in musical instruments or in our use 
of them. The instruments used today are largely the result of 
a gradual development over many centuries. They are not the 
result of scientific research, but of countless experiments in 
making sounds in every possible manner. Only with the advent 
of the telephone, radio loud speakers, the location of airplanes 
and guns in war, and, in general, all forms of acoustic reproduc- 
tion, reception and transmission, has scientific research made 
invaluable contributions to development in acoustics. Today 
money expended annually in research in this one field probably 
exceeds half a million dollars. By the extension of scientific 
knowledge and its application many results have been achieved 
which would have been unattainable by a straightforward, ex- 
perimental, trial and error method. 

The applications have been made in a carefully reasoned 
manner. This has usually required the aid of mathematics not 
readily understood by the novice. The important fact for the 
student to bear in mind is that progress has depended generally 
not upon chance discoveries, but upon the most careful use of 
reasoning power. When phenomena occur that appear inex- 
plicable, we lack either the necessary information concerning 
them, or the ability to reason correctly. For example, one might 
surmise that if speech could be heard and understood from a 
boat to shore, then speech should travel equally well in the re- 
verse direction and should be heard just as easily from shore to 
boat. This surmise arises from many common experiences in 
calling to one another across open spaces or in a building, and 
also from what might be termed "common sense." But these 


do not suffice to give the correct answer, for the transmission is 
not the same in both directions. The wise way to proceed to an 
understanding of the transmission of sound from boat to shore 
is to study carefully what is actually involved in sound trans- 
mission and in its alteration in any manner. Not until these are 
clear in mind can the student consider with profit the transmission 
of sound in this case. This relatively simple problem is illus- 
trative of the necessary procedure. When a problem is compli- 
cated, like the perfect reproduction of all sounds in a talking 
picture, the attitude of mind should be precisely the same. A 
careful understanding of all the phenomena involved, and logical 
reasoning ba,sed thereon, are essential. In the reasoning, mathe- 
matics is indispensable. The well-known processes that have 
been demonstrated in mathematics are capable not only of saving 
the human mind much complicated thought, but also of arriving 
at correct conclusions even in cases where the mind alone could 
not secure a solution. But, in the more simple situations in 
acoustics, the mind can follow the reasoning without the use of 
detailed mathematics. In such cases, it is all the more important 
that the phenomena involved be understood with great clearness. 
It is because of this fact that, in this text, emphasis will be placed 
upon clear understanding. The language must be concise and 
accurate and the reader because of this fact must follow no more 
rapidly than he can comprehend. The meagerness of the mathe- 
matics used is an advantage to the nonmathematical student, 
but this omission thrusts upon him a correspondingly greater care 
in understanding and reasoning. Every aspect considered in the 
early part of the text will be used repeatedly. Therefore, full 
comprehension must be obtained as one progresses. Acoustics 
is a particularly satisfying branch of physics because one can 
visualize the details of the phenomena which are concerned with 
matter and with vibrations therein. Both of these seem con- 
crete and understandable. 

A clear knowledge of the elements of acoustics is becoming 
increasingly important to any profession depending in any man- 
ner upon acoustics. Civilization is becoming acoustically con- 


scious. It is studying the effects of noise. It is more critical of 
all acoustic effects whether in speech, music or sound trans- 
mission. The phonograph of yesterday will not be tolerated 
today. Auditoriums, music halls, and studio rooms must possess 
proper acoustic qualities. Acoustic effects satisfactory in the 
past will not be permitted in the future. Acoustics is an old 
subject, but with new responsibilities of everyday importance. 

1.2. Waves. — The most fundamental concept to be grasped 
is the nature of sound itself. It is said to travel in waves. It 
actually consists of waves in matter. Our long familiarity with 
listening to all manner of sounds does not help us to understand 
wlfat is meant by a sound wave. 

Everyone has witnessed the movement of water waves and 
has recognized that they have a definite speed. A study of water 
waves discloses that the water itself moves neither horizontally 
forward with the velocity of the wave just mentioned, nor ver- 
tically upward and downward. Yet these are the two move- 
ments one recognizes visually as the most likely. It is found that 
any particular portion of the water itself has an approximately 
circular motion,* the plane of the circle being vertical and extend- 
ing in the direction of motion of the wave. If this is the case, 
then if one says the wave has a definite velocity forward he does 
not refer to the water itself but rather to the physical shape of 
the surface of the water. This shape certainly moves with a 
horizontal velocity. To repeat, the term " wave " is used to refer 
to the physical shape and not to any portion of the water itself. 
But carry the ordinary use of the term "wave" a little further. 
The Weather Bureau announces that a "cold wave" is coming. 
Everyone understands that he may expect the thermometer to 
fall. It is a wave of low temperature and is said to be a wave 
because this physical condition has a velocity in a definite direc- 
tion across the country. No one thinks that there is an actual 
movement of the same cold air from one point of the country to 

* Reference is made to the case of waves that are not too great in magnitude 
and that are in deep water. 


another. It is the movement of a physical condition. In a 
similar manner we may have a wave of atmospheric pressure. 

It is noticed that in the above the word "wave" has acquired 
a definite technical meaning. When once clearly understood, no 
further difficulty is experienced. There are found numerous 
physical changes that have at any point a directional velocity. 
These alterations in physical relationships which are propagated 
through the medium concerned with a definite velocity are usu- 
ally referred to as "waves." 

1.3. Properties of Waves. — The water waves mentioned in 
the previous section are caused by the action of gravity, and are 
known as "gravity waves." They are quite different from what 
are termed "ripple waves." The latter are caused by a curious 
surface film of oriented molecules which acts very much like a 
very thin stretched membrane. If the ripple waves are very 
small the motion of the water is vertically upward and downward. 
Thus very small waves and the large wind waves on water are 
different in detail, for the active agency in the propagation of 
the one is the tension or pull in the surface film, and the other, 
the gravitational attraction of the earth. This section will not 
be concerned with this difference, but rather with the use of 
ripples as an illustration of the action of waves in certain respects. 
Figure 1.1 * is an instantaneous photograph taken of a series of 
such ripples on a water surface. They were produced by the con- 
tinuous vibration of a thin wire projecting into the surface and 
vibrating perpendicular to it. This vibrator is located at the 
center of the concentric rings. The surface is brightly illumi- 
nated and the photograph shows the condition of the surface at 
one instant. There are two facts to be noticed. First, the waves 
form concentric circles. Evidently the different parts of any one 
wave have travelled equal radial distances in the same time inter- 
val. That is, the different parts of the wave have had the same 
speed of propagation at the same time. Is there any evidence 

* The photographs shown in Figs. 1. 1 to 1.6 are reproduced by the permission 
of the publishers of "Einfiihrung in die Mechanik und Akustik" by R. W. Pohl, 
Julius Springer, Berlin, 1930. 


that this speed has remained constant as the wave has expanded? 
The answer can be found in reviewing the conditions of the ex- 
periment. The vibrator is in continuous oscillation. This means 
that its frequency y or the number of vibrations per second, is 
constant. Hence, the radial distance on the surface between one 
expanding wave and the one just preceding or following is the 

Fig. i.i 

Fig. 1.2 

distance travelled by the wave in one complete oscillation of the 
vibrator. It is noticed that these radial distances are every- 
where equal. This indicates that the radial speed of propagation 
is everywhere the same. Clearly, then, a single wave maintains 
the same radial speed as it expands. Concerning all the waves 
discussed in this text a similar statement can be made. The 
two facts to be gained by Fig. 1.1 are that the speed of propaga- 
tion of a wave is the same along the different radii, and that it 
continues the same at all times as the wave expands. 

Figure 1.2 shows the ripple waves impinging upon a hole 
made in a wall placed in the water. If these ripple waves trav- 
elled in straight lines as light approximately does, all the waves 
on the far side of the hole would be confined to the angle between 
the dotted lines. The conclusion is that ripple waves do not 
travel in paths which can be represented by straight lines or 
rays, but, on the contrary, have a distinct tendency to spread. 
In fact, according to Figs. 1.3 and 1.4, as the hole is made smaller 


the ripple waves seem more and more to have their origin in the 
hole itself. 

The spreading of the waves in Fig. 1.2 may be described as 
bending around the edges of the hole. This would lead one to ex- 

|^3t;'*^^^*w« : JW^,,^,: 


^^^^f0'^^AAi^'?;QS.' ! Miil&^i 





• '*■ 

Fio. 1.3 

Fig. 1.4 

pect the effect shown in Fig. 1.5. Here the bending prevents the 

obstacle from casting what might be termed a "sharp shadow." 

What occurs at reflection is shown in Fig. 1.6. Here there 

seems to be a new set of waves issuing from the reflector. These 

Fig. 1.6 

are again circular, having a center as far behind the reflector as 
is the original source of the waves in front. A more definite 
understanding of the geometrical symmetry of the location of 
these two centers can be obtained from the discussion of reflection 
of acoustic waves in Section 2.1. 


These ripple waves, with the properties shown, are fairly 
satisfactory as an analogy to sound waves. When one first be- 
comes interested in the physics of acoustics he has need for some- 
thing concrete, even though not strictly like sound waves. The 
properties of the ripple waves as described are similar to those of 
acoustic waves, but the nature of the waves themselves is entirely 
different. But this need not prevent one from gaining a better 
conception of acoustic waves by studying the properties of ripple 
waves. We see that the waves are not propagated in fairly 
straight * lines like light, and that they are not reflected as would 
be a tennis ball from a wall. Indeed, as shown by the action of 
the small hole in a wall, they spread out in all directions from a 
point on the wave. The properties of acoustic waves must be 
appreciated at the outset, though of course the reader cannot 
visualize in detail an acoustic wave. As he becomes more familiar 
with the properties of such waves there will be less need for 
analogies and visualization. The next section introduces the 
reader to the nature of acoustic waves by the consideration of 
waves travelling in but one direction. From the propagation in 
one direction will be developed the phenomena occurring when 
the observer is near the source, and the sound, travelling radially, 
has a different direction of propagation at different points. 

1.4. A " Wave " in a Helix. — If one considers a helix of wire 
suspended as shown in the accompanying Fig. 1.7 he sees that 
it is possible to produce a wave. For if several of the turns at 
one end are pressed together and then the inside turn is released, 
the compressed part will at once expand, producing compression 
ahead. Thus a "wave of compression" will travel along the 
helix, having a velocity or speed of propagation f that depends 
upon the dimensions of the wire and helix and the physical nature 
of the material in the wire. 

* Light does not travel in lines that are exactly straight, but nearly so. For 
our present purpose the straight line propagation of light will be assumed. 

t "Velocity" is the distance passed over per unit of time. Technically it dif- 
fers from "speed" in that the former includes the direction as well as the amount 
or magnitude. 


1.5. Different Aspects of a Wave. — A wave of compression 
has just been described. What other physical alterations are 
there in this wave? First it is to be observed that a wave of 
compression is always accompanied by a wave of rarefaction. 

Fig. 1.7 

For, consider a wave that has been established by giving the 
helix a sudden compression at the end and then restoring that 
end to its original position of rest. Suppose the compression to 
be travelling along the helix, both ends of which are in their 
original undisturbed positions. The actual length of the helix is 
unaltered, although there is a portion where the helix is com- 
pressed. If the total length is unaltered there must be a portion 
where the helix is elongated. So a wave of compression in the 
helix must be associated with a wave of rarefaction. These are 
indeed two aspects of the same wave. Again, one can observe 
in this experiment that the progress of the wave will cause any 
given turn to oscillate or vibrate to and fro during the passage 
of the wave at that point. This vibration is readily shown to the 
eye by tying a bit of string at the bottom of one of the turns of 
wire. Each and every point of the helix suffers a displacement * 
from its position at rest, and this displacement is first in one 
direction and then in another. We may say that a wave of dis- 
placement has travelled along the helix. In fact, it can be 

* The word "displacement " is sufficiently defined by its use here. The amount 
or magnitude of the displacement is the actual distance from its position of rest. 


shown * that a wave of displacement and a wave of compression 
not only are, but must be, coexistent. 

The fourth aspect of the helix wave is somewhat more ab- 
stract. Velocity is the rate of movement in a given direction. 
It is the distance covered per unit of time. As a turn of the helix 
is displaced, it has a definite velocity in that direction. As the 
wave of displacement passes, the helix experiences a velocity (not 
constant or uniform) first in one direction and then in another. 
We could consider this aspect of the wave and call it a wave of 
impressed velocity. But one can see at a glance that the velocity 
referred to is not the velocity of the wave, but of a given portion of 
the helix. There is then a very definite distinction between the 
velocity of the wave described in Section 1.4 and a wave of velocity 
considered in this section. This is true of the helix and it is also 
true of a sound wave. 

As above shown, there are four different aspects of this wave 
in a helix, compression, elongation, displacement and velocity. 
There is an analogy in the acoustic wave in a gas. A gas resists 
compression and will return to its former volume after compres- 
sion (cf. automobile tire). So does the helix. A gas has inertia, 
that is, time is required to set it in motion. This is true of the 
helix. Indeed, these two qualities, which are called "elasticity" 
and "inertia," make possible the existence of a wave and its 
movement at a definite speed. It is sufficient for the present 
purpose to check up this thought with the helix. The wave of 
compression moves forward because the compression at one point 
exerts a force attempting to compress another point just in front. 
Without elasticity this would not occur. Moreover, if the helix 
were without inertia, any force would produce an effect instantly. 
This would result in an infinite velocity of the wave. This is, of 
course, not imaginable because we have never experienced any- 
thing without inertia and yet with ability to exert a force. A 
gas, having these two qualities, elasticity of compression and 
inertia, will transmit a wave of compression with a definite veloc- 

* The phrase "it can be shown " will be frequently used. It refers to a possible 
demonstration but not one proposed for the student. 


ity of propagation. Such a wave is called an acoustic or sound 
wave. For the purposes of this book, the wave will be so desig- 
nated even if it cannot be detected by the human ear. 

1.6. Gas as a Medium for Sound Waves. — That substance 
which transmits a sound wave is called a "medium" or means of 
transmission. In the case of the helix the medium is a continu- 
ous one, that is, the helix is a continuous piece of wire. Upon 
closer examination the wire is found to be made of fragments of 
crystals and in each of these fragments an orderly array of atoms, 
separated from each other, but nevertheless exerting forces upon 
one another. A force is required to compress or elongate such a 
crystal in any way. But for our present purpose we will not 
inquire as to what these individual atoms and crystals are doing. 
We will not go further than to appreciate the four aspects of the 
wave in the helix as previously described. In a similar manner 
we will neglect any consideration of the molecular constitution of 
a gas. The separation of these molecules is, on the average, very 
much greater than the diameter of one of them. To add to the 
detail, it should be stated that these molecules are moving to and 
fro in every direction, these motions corresponding to the heat 
the gas possesses. We are to be satisfied with the fact that the 
gas has elasticity and inertia, just as if it were continuous, and 
therefore it will act acoustically as an imagined continuous me- 
dium. No reference will need to be made to the motions of the 
individual molecules. Hereinafter when the phrase "a particle 
of the medium" is used it is understood that this does not refer 
to a molecule, but to a small portion, called a "particle," of the 
medium imagined to be continuous. This substitution of imag- 
ined continuity will be a great convenience. 

1.7. Representation of a Sound Wave. — A sound wave may 
occur in a solid, a liquid or a gas. But in the last two, or fluids, 
the wave takes only the form of a wave of condensation and rare- 
faction, as already described. The discussion in this text is lim- 
ited practically to that kind of a wave. In -the case of most 
vibratory bodies producing aerial sound waves, such as a vibrating 



string or a vibrating air column in a wind musical instrument, 
the waves of compression and rarefaction produced in the air 
and reaching the ear follow one another in succession, but with 
the pressure * of the air at any point changing with time in an 
interesting manner. One may graph a series of such waves of 
changing pressures as in Fig. 1.8. Here the changing value of 
the pressure at a point O is represented by distances above and 
below the horizontal line, each positive and negative value being 
respectively an excess and a shortage of pressure when compared 
with the mean pressure. Figure 1.8 as drawn actually represents 

a type of variation with time that has been found to be the sim- 
plest in every respect. This variation may be that of a quantity 
like pressure, but it also may be one of displacement from a mean 
position. It is, in fact, approximately the variation in displace- 
ment with time possessed by a vibrating pendulum. On first 
thought one would scarcely refer to the motion of a pendulum 
as simple, for its velocity changes constantly as it swings from 
its mean to its extreme positions. It might seem that a simpler 
motion would be one of uniform speed everywhere except at the 
ends of the arc where the pendulum could be stopped suddenly. 
But this is not the case, because two entirely different kinds of 
motion are assumed, one a uniform velocity and another an 
abruptly changing velocity. Being so different they could not 
be simultaneously described in a simple way. But with the pen- 
dulum the velocity changes progressively in a manner that per 
mits a relatively simple specification. In its most condensed 

* "Pressure" in a gas technically means the force per unit area which the gas 
would exert upon any containing wall. 



form this statement is entirely mathematical. But a visual de- 
scription can be obtained by observing certain actual motions. 
The three driving wheels on pne side of a steam locomotive are 
connected and driven by a 'side or parallel rod which in turn is 
attached to the piston. At each wheel the side rod bears upon 
a crank pin which is made a part of the wheel itself. Assume 
that these wheels are being driven by the side rod, but that they 
are slipping on the track without any forward motion of the 
engine. Let the observer, who is standing alongside, fix his atten- 
tion upon the pin on the wheel. It spins about in a circular 
motion. Suppose, however, that the observer were standing at 
a distance of perhaps fifty feet from the engine and yet alongside 
the track. Suppose also that although he is standing practically 
in the plane of the revolving driving wheel, he is yet able to see 
the moving end of the side rod or the crank pin itself. He will 
observe now not a circular motion of the pin but one upward and 
downward, with the entire movement appearing to occur in prac- 
tically a straight line. In this apparent motion, the velocity of 
the pin will vary, being maximum at the center of its vertical 
path and zero at the extremities. 

In the lecture room an experiment can be arranged as follows. 
A horizontal beam of light issues from a lantern and falls perpen- 
dicularly upon the screen. In its path is placed a bicycle wheel 
rotating at constant speed about a vertical axis. The shadow of 
the wheel rim or tire appears as a horizontal line. If a ball is 
fastened above the wheel and at the rim, its shadow remains on 
the screen during the rotation of the wheel. This shadow of the 
ball moves to and fro, apparently in a horizontal straight line. As 
will be surmised from an earlier statement this motion of the 
shadow has the same kind of varying velocity as that of a pendu- 
lum ball. The motion of the shadow is also a visual description 
of a pendulum's motion. Moreover, this description is simple, for 
it involves only a projection of a uniform motion in a circle, in 
itself a very simple kind of motion. It is not surprising to learn 
that when the motion of the shadow of the ball is described 
mathematically it proves to be very simple indeed. But it is now 


necessary to revise the conditions of the experiment to conform 
exactly with the mathematical statement mentioned. The light 
falling on the ball must be parallel and not at all divergent. Then 
the shadow has exactly the same dimensions as the wheel itself. 
With this alteration it can now be said that the motion of the ball's 
shadow is the kind of vibration or oscillation used throughout the 
subject of acoustics. It is called "simple harmonic motion," and 
is the simplest type known. Clearly one may similarly refer to a 
simple harmonic variation of pressure, or any other quantity, 
when its variation is like that of the displacement of the shadow of 
the ball from a mean position. Now it happens that the simplest 
vibration made by a tuning fork, by a piano string or by almost 
any vibrating mechanism, is precisely of the same character. A 
simple musical tone is a simple harmonic variation of pressure 
such as indicated in Fig. 1.8. The usual musical tone consists 
of a number of such simple tones. In the foregoing discussion 
the motion of a pendulum was selected because it is familiar. It 
must now be admitted that its motion is not strictly simple har- 
monic, as is that of the shadow of the ball moving in a circle, 
but approximates very closely to that condition. On the other 
hand, a simple pure tone does consist of a simple harmonic varia- 
tion of pressure as stated. Moreover, the variations of displace- 
ment and particle velocity are also simple harmonic. This type 
of variation or vibration is the kind with which we are particu- 
larly concerned in acoustics. It is the building unit out of which 
we will construct or describe complex tones. 

It has just been stated that the time variation of the excess 
pressure at a point in the medium can be represented graphically 
by a continuous curve. This is a wave of pressure. One could 
however, represent the same kind of a wave by considering a row 
of little masses joined by an idealized weightless elastic cord as 
in line A of Fig. 1.9. Let there be a compressional wave along 
the cord similar to the one previously in the helix. Then line B 
will represent what is happening at a certain instant to the row 
of masses. 

As the wave travels, for example, from left to right (just as 



in the helix), the masses suffer a to and fro displacement along 
this horizontal line. It is not possible, without confusion, to 
represent the displacements of all these masses by drawing lines 
in their actual directions, for these would all lie in the same straight 
line, i.e., the direction of the cord. But the displacements can 
be represented clearly by selecting a somewhat arbitrary method. 
If, from the mean positions of the particles, lines proportional 
but perpendicular to this actual displacement are drawn, a curve 
through the ends of these lines may be said to represent the dis- 
placements at that given instant. For example, the undisturbed 


Fig. 1.9 

A. An elastic cord with equidistant loading. 

B. A compression al wave in the cord. 

C. A graph showing displacements from mean positions of these loads. 

position of the small masses is shown in A. But if a longitudinal 
wave is passing along this row of masses, then, at a chosen instant ', 
the position of these same masses may be indicated by the draw- 
ing in B. If the actual horizontal displacement of a given mass, 
as shown by a comparison of A and B, is represented by a vertical 
line of the same length, perpendicular to the horizontal line in C, 
but drawn from the undisturbed position as in A, and if this rep- 
resentation is repeated for each particle, a curve drawn through 
the ends of these lines may be regarded as representing the dis- 
placements of the masses at the given instant. Displacements 
to the left and right have been represented by displacements 
down and up, respectively. This is an awkward method, because 
the graph does not truly represent the direction of a displace- 
ment. The graph in C nevertheless is said to represent the dis- 
placements of the row of masses at a given instant.* The masses 

* The solid circles on curve C represent the pseudo-positions of the masses, 
the positions they would occupy had the displacement been up and down. 


and the elastic cord can now be replaced in imagination by a 
gaseous medium, and Fig. 1.9 then gives a visualization of what 
is transpiring in a compressional wave in a gas. The graph of 
displacement is to be regarded as referring to the "particles" of 
the medium. Curve C in Fig. 1.9 is then a representation of a 
compressional or sound wave in a gas at any instant. If the 
wave can be considered as moving from the left to the right 
with a definite speed, one can prophesy just what will happen 
to a given particle at a certain time. 

In curve C, the vertical distance above the horizontal line 
represents a displacement to the right, and the one below or 
downward, a displacement to the left. Having in mind the cor- 
respondence between the displacement to the right and the rep- 
resentation drawn upward, it is observed that the displacements 
in the neighborhood of the point "«" are directed toward that 
point from both sides. Consequently " a " is a point of maximum 
pressure at that instant. By similar reasoning " b" is observed 
to be the point of minimum pressure at that instant. There are 
two differences in the graphs in Figs. 1.8 and 1.9. Not only do 
they refer to variations of different quantities, but the former 
refers to the variation at a point as time elapses, while the latter 
is an instantaneous picture, so to speak, of the condition of a 
row of particles lying in the direction of the passage of the wave. 
There is a certain similarity between Fig. 1.9 and a gravity wave 
at the surface of water, though, in point of fact, the two are not 
of the same shape. The "crest" of a water wave is not the same 
shape as the "trough" and this is caused by the fact that the 
motion of the water particles is circular. 

1.8. Velocity. — As already suggested by a statement in re- 
gard to the helix, the velocity of a wave depends upon the medium 
in which it is propagated. It is possible to prove that in a gas 
the velocity of a sound wave is 

v=<\\k£> (I.I) 


where p and p are the undisturbed pressure and density * under 
normal conditions, respectively, and k is a quantity depending 
upon what are called the specific heats f of a gas. While it is 
not proposed to examine the reasons for the form of this equation, 
we can with some satisfaction notice that the equation is in 
accord with the following considerations. A gas at high pressure 
recovers from a compression quickly just as a stiff spring under 
high pressure will, if released, return to its original position speed- 
ily. It appears reasonable that a quick recovery would result 
in a high velocity of the sound wave, and the formula (i.i) states 
that v is proportional to Vp. But if one increases the mass of 
the spring, the recovery cannot be so rapid, for with the same 
force acting, the more massive the body the more slowly it can 
be set in motion. The formula states that the greater the den- 
sity, pressure remaining constant, the less the velocity. This 
seems to be in accord with the variation just suggested. 

A definition of "velocity of a sound wave" has been inferred. 
It is the speed with which the physical alteration is propagated 
in a definite direction. For example, if one represents a maxi- 
mum displacement as at the point midway between b and a in 
Fig. 1.9, and if the wave is moving to the right, the velocity is 
the speed to the right which one must travel in order always to 
be at this point of maximum displacement. 

It is necessary to refer to the velocity of sound in liquids and 
solids, for acoustic waves can be transmitted in any material. 
Formula (1.1) applies to gases only. In liquids it is customary 
to employ a slightly different form, as follows: 


Here £ is a symbol representing what is technically called the 
volume elasticity of the medium. The reader will not be asked 
to become familiar with this technical definition, but merely with 

* "Density" is the amount of the gas per ccm. Technically, it is measured 
in mass per ccm. or grams per ccm. 

t The exact definitions of "specific heats " need not concern the reader. Obvi- 
ously heat enters into the situation, for when a gas is compressed it is heated thereby 
and when expanded it is cooled. 



the statement that volume elasticity is a measure of the force 
required to reduce the volume of the material by a fixed amount. 
The value E must be obtained for each liquid, so that there is no 
way in which one can compute the velocity in one liquid from 
the knowledge of the velocity in another. In gases, there is a 
slight variation in "k" but this variation is known to be caused 
by differences in the number of atoms in a molecule. 

It is evident, however, that the nature of the vibrations in 
gases and liquids is the same. In both, molecules can change 
positions relative to one another freely, but in both there is an 
opposition to compression or expansion. In a solid, the mole- 
cules are close together and are fixed in relative positions. It 
will resist compression and expansion and there is possible the 
propagation of an acoustic wave in a given direction in the same 
manner as in a gas or liquid. There is also in a solid resisting 

Table I 



Velocity in meters 
per second 





Hebb (1919) 






! o° 


Reid (1930) 

Carbon dioxide 

Atmospheric press. 


257 to 260 

Various observers 


Atmospheric press. 


1238 to 1269 

<« u 


Atmospheric press. 



« « 

Water distilled 



Dorsing (1908) 










Eckhardt (1924) 




Masson (1858) 




Wertheim (1849) 




Cast steel 




<« M 








Hard rubber 



Stefan (1872) 

Vulcanized rubber 

5o°-7°° ; 


Exner (1874) 


. — 


Kundt (1868) 


force to other kinds of motion such as a twist. Consequently, 
the transmission of sound in solids is more complicated than that 
in gases and liquids. In a wave of compression such as already 
discussed, the vibrations are in the same direction as the progress 
of the wave. Such vibrations are called technically "longitu- 
dinal." At a later point reference will be made to waves other 
than longitudinal that can be transmitted by a solid. Until 
then, any mention of sound transmission in solids will refer only 
to longitudinal waves. 

The velocity of a sound wave is not wholly independent of 
the nature of the wave, as is evidenced by experiments with ex- 
plosions wherein velocities have been found considerably in excess 
of the normal sound velocity. This is discussed in Section 1.9 
but practically all sounds herein discussed travel with the same 
velocity, called the normal velocity. The accompanying Table 
I gives selected values. 

1.9. A Variation of Velocity. — It will be noticed in (1.1) that 

we have \jk-- If a gas is compressed so that p y the pressure, is 

doubled, it is found by experiment that p is doubled also. Hence 

the ratio of the two remains the same and consequently (1.1) states 

that there is no alteration in the velocity of sound. The same 

conclusion would be reached for an expansion of the gas. But this 

constancy of the ratio, with pressure changing, is true only if the 

temperature remains constant. For example, the temperature of 

a gas may be increased by heating, keeping the volume and hence 

the density constant and yet increasing the pressure. In fact, 

the ratio between p and p is determined by the temperature. 

Therefore the velocity of sound in a given gas depends only upon 

the temperature and not at all upon the values of pressure and 

density. It can be shown both experimentally and theoretically 

that the velocity is proportional to the square root of the absolute* 

* Degrees on the absolute scale are very nearly the same size as on the Centi- 
grade scale, but o° C. is 273 absolute. Hence we generally add 273 to the reading 
of the Centigrade scale to get the absolute temperature. In the Centigrade scale 
water freezes at o° and boils at ioo ', these temperatures being written o° C. and 


temperature. Hen ce at o° C. for i°C. rise in temperature the 
velocity must be ^J 2 * 73 + * times the velocity at o° C, and at 

/° C. it must bey j 273 +/ times the velocity at o° C. or 

v t = v Q yj 1 + — = t; V(i + .00366/), (1.3) 

where Vq is the velocity at o° C. and v t at t° C. 

If a gas is under very great pressure, an exception must be 
made to the statements in the preceding paragraph. While it 
is true as stated that for a change of pressure under ordinary 
conditions the velocity changes but a negligible amount, yet at 
high pressure there is a marked change, especially at low tem- 
perature. This change is not only one of magnitude but is some- 
times positive and sometimes negative. At — 103. 5 C, the 
velocity with one hundred times the atmospheric pressure, or 
"100 atmospheres/' is, according to Koch (1908), 293.2 meters 
per second, while with 150 atmospheres it is 346.9 and with 200 
atmospheres it is 406.5 meters per second. Witkowski (1899) 
found that, at the same temperature, the velocity decreased from 
260 to 245 meters per second when the pressure was increased 
from one to forty atmospheres. 

The velocity in free air will change with humidity, because 
at the same pressure the presence of water vapor will alter the 
density. But this change is always less than one per cent if the 
atmosphere is saturated with moisture at ordinary temperatures. 

Yet another exception will need to be made to the statement 
that the velocity of a wave in air depends only on the tempera- 
ture. In the mathematical study of the passage of waves of 
condensation and rarefaction in a fluid, it has been proved that 
the velocity is independent of the magnitude of the displacements, 
but only if these are small. It might be anticipated, therefore, 
that waves of abnormally high velocities can be produced. This 
has been repeatedly accomplished by explosions. Even the waves 
near a large gun travel at a higher speed than the normal ones. 


A review of earlier experiments is described by Professor A. L. 
Foley.* The speed of the waves produced by sparks has been 
studied by Foley and reported in the article just cited. He 
found that the speed close to the source depended upon the in- 
tensity and that in the case where he was able to obtain twice 
the normal velocity at a distance of 0.32 cm. from the source, 
the velocity had decreased to the normal value at 2 cm. from 
the source. 

1. 10. Frequency and Wave-Length. — The frequency of a 
sound vibration is usually the number of complete, or double, 
vibrations of the particles of the medium per second. The fre- 
quency is sometimes designated in "cycles." Yet some manufac- 
turers mark on their tuning forks the number of single vibrations. 
The pitch of a musical sound is determined by the frequency. 
The higher the frequency the higher the pitch. The wave-length 
is the distance the sound travels during the time of one complete 
vibration. Thus the distance travelled in one second would be 
the length of one wave repeated that number of times which 
corresponds to the frequency. Hence the relationship between 
velocity, frequency and wave-length is as follows: 

Velocity = frequency X wave-length. (1.4) 

In passing from one medium to another the frequency is con- 
stant, for adjacent particles at the boundary of the two media 
must vibrate together. But the velocity of the wave is not in 
general the same in the two media. If the frequency is constant 
but the velocity different, the wave-length must also be different. 

1.1 1. Doppler's Principle. — Reference will now be made to 
a common phenomenon. If the source is in motion in the me- 
dium in a certain direction, then, though the frequency of the 
source remains unchanged, the wave-length measured in the 
medium will be altered. On the side of the source which is in 
the direction of motion the wave-length will become shorter and 
on the other side longer. To a stationary auditor, standing near 

* Foley, Physical Review \ 16, p. 449 (1920). 



the path of motion, the frequency at the approach of the source 
will be greater than at the recession. This is a common experi- 
ence with a train whistle or an automobile horn. It is not difficult 
to see that frequency heard will depend upon the auditor's veloc- 
ity relative to the medium also. Illustrations of both aspects are 
left to the reader. 

1. 12. Velocity of the " Particle " of the Medium. — It is easy 
to confuse the velocity of sound with the velocity of a particle 
in the medium. The former is large, as has been shown in Table 
i.i, but the latter is small. The reason is readily appreciated 
if one will notice that the maximum displacement in a sound 
wave is an exceedingly minute fraction of a millimeter. Even 
though the particle oscillates several thousand times a second, it 
can be shown that the velocity at its mean position is usually of 
the order of a small fraction of a millimeter per second. Thus 
these two velocities mentioned above differ not only in meaning 
but enormously in magnitude. The velocity of the wave is not 
the velocity of the particle. 


i. Why is a "displacement" necessary in a "wave of condensa- 

2. Indicate in Fig. 1.9 the positions where the following conditions 

{a) The displacement is a maximum to the right, to the left. 
{b) The displacement is approximately the same for neighboring par- 
(c) The displacements differ the most widely for neighboring particles. 

3. Show that the maximum pressure and maximum displacement 
at a given point do not occur simultaneously in a sound wave. 

4. An organ pipe changes its frequency with temperature because 
(as hereinafter shown) its length (assumed to change inappreciably 
with temperature) must remain one-fourth of the length of the wave. 
What will be the change in frequency if the room is heated from 
o° C, where the frequency is 256 per second, to 20 C. ? 

5. Will the change in velocity of a sound wave in air caused by a 
change of temperature have any direct influence on the pitch of a piano 
or violin? Explain. , 


6. A water wave and a sound wave in the water travel horizon- 
tally to the right. What difference can you point out in the vibra- 
tions of the water in the two waves? 

7. In what respect does the following expression seem misleading: 
"Sound travels in waves"? 

8. Assume in Fig. 1.8 that the wave is travelling to the right. If 
you had an instantaneous picture (drawing) of the wave of pressure 
in a row of particles, how would it differ from the drawing of Fig. 1.8? 
Consider the drawing to represent the condition at the time indicated 
by o in Fig. 1.8. 

9. What is the difference between the velocity of a sound wave and 
the velocity of a particle of the medium? 

10. According to Eq. (1.1), what can produce a change in v if the 
gas concerned is open to the atmosphere? 

11. Both thep and p of the atmosphere change from time to time. 
How would you determine the velocity at any one time if you knew 
the velocity for at least one condition of pressure, density and tem- 

(It is suggested that a review of Chapter I be made before pro- 
ceeding with Chapter II since a clear understanding of these funda- 
mental concepts is necessary for the comprehension of the remainder 
of the text.) 



2.1. Reflection at a Plane Surface. — Sound is not reflected 
like light, or like a tennis ball. It is a wave of pressure and of 
displacement. The purpose of this section is to describe what 
occurs as a result of reflection and that without the use of any 
analogy. Take a specific case. A source of sound is placed at 
0, Fig. 2.1, in front of a wall, represented in cross section by a 
line. It is desired to ascertain the effect of the wall upon the 
incident sound. At the wall at a point between and 0', which 
is the same distance from the wall on the other side, the sound 
from impinges upon the wall in a direction perpendicular 
thereto. At no other point along the wall does 
this perpendicularity exist. But a study of the 
reflection of a sound wave must include what 
occurs at all points of the wall. Obviously the / 
situation is quite complicated and hence the physi- V R 4 
cal action at the wall is perhaps too difficult to >y / 

describe in detail. So the physicist seeks an in- 
direct method, which, as will be found, permits # o' 
him to describe the reflected wave without the 
necessity of detailing the action of the reflection 
over the entire wall. Let an imagined source, Fig. 2.1 
0', like that at 0, be placed on an extended line 
drawn from perpendicular to the wall, the distance of and 0' 
from the wall being the same. By the method referred to, it will 
be shown that the sound coming from will be reflected from 
the wall in such a manner as to produce a reflected wave precisely 
like that which would have come from a similar source, placed, 
at 0', if the wall were absent. This is a remarkably simple de- 
scription of a complicated physical action. It is interesting and 
instructive to follow the demonstration of its truth. 


R X 



Assume the sources and 0' as stated, but at first without the 
wall. The waves emitted by and 0' are spherical. Select any 
point P on the plane between and 0' and consider the sound 
wave arriving from at the time it has a positive displacement. 
But if 0' is a like source, then there is arriving simultaneously a 
wave from 0' with a positive displacement of the same magni- 
tude. Since a positive displacement is in the direction of the 
wave motion, the arrows as shown give the correct directions OP 
and O'P of the displacements. 

It is now necessary to consider how one can ascertain the 
resulting displacement. A well-known method is to complete the 
parallelogram with the two arrows representing the given dis- 
placements as sides and to regard the diagonal as the resultant R. 
This method, while apparently reasonable, is not easy to prove 
in a few words, and hence will be assumed. 

Applying the foregoing to the two equal displacements at the 
point P, with the wall absent, the resulting displacement, R y is 
clearly in the plane which indicates the position of the wall when 
present. Since the resultant is the displacement which actually 
occurs, this means that there is a displacement in the plane 
mentioned but none perpendicular to this plane. But P is any 
point in the plane, and hence the condition of zero displacement 
just stated is true everywhere. This is true at the instant chosen; 
it will be true at any other instant since the two component dis- 
placements are always equal and always make equal angles with 
the plane. Inasmuch as the displacement and hence the motion 
perpendicular to the plane at any point are always zero, we can 
substitute a real motionless wall for the imaginary plane without 
modifying the resulting wave motion on the right of that plane. 
For with the wall present and assumed to be rigid, we can have 
no motion perpendicular to the plane. This is precisely the same 
condition stated above when the wall is absent and there are the 
two like sources and 0'. Thus with the source 0, the rigid 
wall and the reflected wave, the resulting condition on the right 
is equivalent to the two sources O and 0' without the wall. 
Hence the reflected wave from the wall is the same as if it origi- 


nated at 0', which is called the image of 0. In Fig. 1.6 was shown 
the reflection of a series of ripple waves from a small plane re- 
flector. Imagine this to be a part of an infinite plane as used in 
Fig. 2.1. Then it is easily seen in Fig. 1.6 that the phantom 
source of the reflected wave, or the image, is as far behind the 
plane containing the reflector as the source is in front, and that 
the two lie on a line perpendicular to that plane. The experiment 
in Fig. 1.6 thus illustrates the discussion in this section. 

It is evident from the preceding that it is not difficult to 
obtain the result of reflection accurately, for the acoustician will 
in his computations treat the effect of the wall as that of an image 
at the point 0' equally distant from the plane. But being able 
to compute the result is not the same as understanding precisely 
what occurs at the reflecting plane itself. If one desires to con- 
template the act of reflection, he should avoid thinking of the 
displacements and consider rather a wave of pressure. Then it 
is easily appreciated that, as a variation of pressure is created at 
the wall, there will be a wave propagated therefrom. 

2.2. Echo Reflecting from a Rough Surface. — The echo as 
commonly known is merely the sound from the "image" we have 
described. In the production of echoes, the best effects are found 
with plane surfaces of considerable dimensions. Nevertheless, a 
rough surface may be used, such as the edge of a grove of trees. 
But the indentations in the plane must be not large in compari- 
son with the wave-length of the sound used 1 , for then the argu- 
ment which has been given in the preceding paragraphs, depend- 
ing upon equality of phase at the surface, will not hold good. 
For exact equality of phase demands that P y which is any point 
in the plane, must be equidistant from and 0'. If the reflector 
is really not a plane, then there can be no single image 0' that 
always is at this prescribed distance from any point P. 

The chief reason for the failure of an echo with a small surface 
is virtually that the sound bends around it, leaving little to be 
reflected. This phenomenon is called diffraction and will meet 
our attention at a later pointy >\ *.., 


2.3. Reverberation. — If a source of sound is placed in a 
room all the walls reflect and consequently make the magnitude 
of the sound greater than if the walls were absent. The waves 
are reflected not once but again and again and it becomes im- 
possible to continue to trace the waves originally sent out from 
the source. If a sustained sound is used, the resulting intensity * 
continues to increase and if there were no absorption of the sound 
energy, there would be no limit to the intensity of sound in the 
room. The absorption is quickly appreciated if the source is 
discontinued. Then it is observed that the intensity does not 
remain constant but decays gradually, evidencing an absorption 
of energy. In a large empty auditorium the residue of sound 
may continue for several seconds. The repeated reflection of 
sound in a room is called "reverberation" and the "time of 
reverberation" is the time required for the sound to become 
inaudible after the source is discontinued. It should be observed 
that reflection occurs from all the objects in the room as well as 
from the walls, ceiling and floor. 

2.4. Absorption. — Usually by " absorption " of sound the 
physicist means the transfer of acoustic energy to heat energy. 
This is occasioned by the fact that if the adjacent portions of a 
gas are compelled to slip past one another heat is developed. 
This property of resistance to "slip" is termed "viscosity." 
There is viscosity in solids and in liquids as well. In considering 
the flow of fluids in a pipe it is customary to assume that the fluid 
at the wall does not move but that the viscosity of the fluid itself 
is the cause of the resistance to flow. It is internal friction. 
Obviously a method of producing sound absorption in a gas is to 
let the sound pass into a large number of small channels where 
the slippage already mentioned will occur. Thus, because of its 
physical construction, a heavy rug will produce absorption. Also, 
it is to be noted that the small fibres of the material will be caused 
to move slightly, though the displacement is a microscopic dis- 
tance, and hence that there will be an additional absorption of 

* "Intensity" refers not to loudness but to the amount of energy per unit 
volume in the sound wave. •' ' '» a 


energy in these vibrations on account of the viscosity of the 
material. These facts will receive attention in a later paragraph. 

2.5. Reverberation in a Room. — The appreciation of the 
significance of the reverberation in a room and the methods of 
diminishing it to a desired degree marked the beginning of the 
scientific study of architectural acoustics. Professor Wallace C. 
Sabine in 1895 De g an tne study of the relation between rever- 
beration and the properties of the materials present in the room. 
He found in existence the widespread belief that the stringing of 
wires greatly assisted in reducing reverberation. Indeed, at that 
time, numerous auditoriums throughout the country were strung 
each with several miles of wire. From a later section devoted 
to resonance the reason for the inadequacy of wires can be ascer- 
tained. Dr. Sabine found very quickly that the time of decay 
of the sound to inaudibility was caused by absorption of all mate- 
rials present. Reverberation is not wholly undesirable. Indeed, 
as previously shown, it augments the intensity. But if the sound 
of one syllable enunciated by a speaker lasts long enough, it will 
interfere with a clear understanding of the succeeding syllable. 
A similar undesirable confusion occurs with music. 

The optimum (or most favorable) time of reverberation must 
be determined by the auditor. An interesting experiment per- 
formed by Dr. Sabine in regard to music rooms will illustrate the 
point. When the New England Conservatory of Music was com- 
pleted, the piano rooms were found quite unsatisfactory. An 
appeal was made to Dr. Sabine, who agreed to undertake an 
investigation. An important part of the problem was to ascer- 
tain the opinion of musical experts in regard to the most desirable 
time of reverberation. A committee consisting of the director 
of the conservatory and four members of the faculty was asked 
to assist in the experiments. The opinion of the committee was 
ascertained by having each member pass judgment upon the 
effect of piano music in the various rooms, the amount of absorb- 
ing material present in each case being made variable by the 
insertion and removal of theater cushions. It was found that 


the committee agreed remarkably well. In fact, to express it in 
the terms of the unit used, they agreed as to the proper effect to 
within one theater cushion. All the rooms were tested, and at a 
later time Dr. Sabine actually measured the time of reverberation 
with the same amount of absorbing material, making due allow- 
ance for the number of persons originally present. The result 
for the optimum time of reverberation for a piano is 1.08 or 
practically 1.1 seconds. The rooms varied in size from one to 
three times. The furniture differed considerably. Yet the fig- 
ures for the various rooms agreed to within the error produced by 
one cushion. One might conclude that the correct time of rever- 
beration for small piano rooms is 1.1 seconds, and that a general 
statement for piano music would require further experiments in 
auditoriums of all sizes. But this conclusion must be modified 
for, in general, the preference for an optimum time depends upon 
the experience of the individual. Even orchestral directors differ 
as to the best time of reverberation. Perhaps 1.0 second for 
small auditoriums and 1.8 seconds for large auditoriums are ap- 
proximately near the satisfactory figures. 

If an effort is made to reduce the measurement of the time 
of reverberation to a basis whereby the time can be computed in 
advance of the construction of an auditorium, one must have a 
standard for a perfect absorber. If sound from within passes 
through an open window, very little is reflected and practically 
all absorbed, that is, never returns. It would be possible then 
to determine the absorbing quality of other materials in terms of 
an open window of the same area. If a piece of hair felt, for 
example, when placed against a plaster wall absorbs 50 per cent 
of the amount of sound energy absorbed by an equal area of open 
window, the "absorption coefficient" of hair felt is 0.50. 

Assume a source of sound emitting acoustic energy at a fixed 
rate. Before any reflections take place, the sound reaching an 
auditor will have the same intensity as were the source and the 
auditor not surrounded by walls and other materials. But the 
waves strike the walls and other objects and these reflections mul- 
tiply; the intensity continues to increase until the absorption 


occurs at the same rate as the emission. There are two factors 
involved in the continuation of the sound after the emission is 
discontinued. First, it is obvious that, since absorption occurs 
at each reflection, the rate at which the sound is absorbed will 
depend upon the number of reflections per second. Inasmuch 
as the velocity of sound is fixed, this then means that the rate 
depends upon the dimensions of the room, for the larger the 
room, the less the number of reflections per second. Hence the 
rate at which the sound is absorbed will decrease as the volume 
of the room is increased. Or the time of reverberation will in- 
crease with the volume of the room. Second, the rate of absorp- 
tion will increase and the time of reverberation will decrease with 
the absorption coefficients of the room. When a careful study- 
is made, it is found that the time of reverberation does depend 
upon the volume, V, and also upon what is termed the "absorb- 
ing power," hereafter referred to as "a" which is the sum of all 
products obtained by multiplying the area of each exposed mate- 
rial by its absorption coefficient. Dr. W. C. Sabine determined 
the time of reverberation experimentally with a certain source 
and found that it was, 


all dimensions * being in meters. 

His experiments showed that (2.1) was true in auditoriums 
usually met in practice. A few years after this first work of Dr. 
Sabine, a theoretical article by Dr. W. S. Franklin,f assuming 
that sound can reach any part of the room with ease and assuming 
a source similar to Sabine's, obtained the following: 

t = 2^Z. (2 . 4) 

The agreement of the experimental equation (2.1), with the the- 

* If all dimensions are in feet, then / = • 


t Physical Review, 1903. 



Table II 



i. Acousti-Celotex, Type A perforated 
fiber board, 13/16" thick, 441 holes 
per sq. ft., 3/16" diameter, 1/2" deep, 
plain side exposed 

2. Akoustolith Tile, 7/8" thick, fine tex- 
ture, cemented to clay tile 

3. Balsam Wool, soft wood fiber, paper 
backing, scrim facing, 1" thick, .254 
pounds/sq. ft 

4. Standard Celotex, 7/16" thick on 1" 

5. Draperies, hung straight, in contact 
with wall, cotton fabric, 10 oz. per sq. 

6. The same, cotton fabric, 14 oz. per 
sq. yd 

7. The same, velour, 1 8 oz. per sq. yd. . . 

8. The same as No. 7, hung 4" from wall 

9. The same as No. 7, hung 8" from wall 

10. Cotton Fabric, 14 oz./sq. yd., draped 
to 7/8 its area 

11. The same as No. 10, draped to 3/4 

12. The same as No. 10, draped to 1/2 

13. Felt, Standard 1", all hair. 

14. Felt, Asbestos-Akoustikos (hair and 
asbestos fiber), 1/2" thick 

15. The same 1" thick 

16. The same 1-1/2" thick 

17. The same 2" thick 

1 8. Flax-linum, semi-stifT flax fiber board, 
1/2" thick 

19. Masonite, Standard 1/2" board 
(pressed wood fiber), laid on 1" fur- 
ring, i8"O.C 

20. 1" Nashkote AAX, 1" felt with cotton 
fabric cemented on surface, two coats, 
special paint 

21. Plaster, gypsum on wood lath on wood 
studs, rough finish 

22. The same with smooth finish ("lime 

23. Plaster, lime on wood lath on wood 
studs, rough finish 

24. The same, smooth finish 

25. Plaster, " Calacoustic," 1/2" thick. . . 

26. Plaster, Sabinite, 1/2" thick 

128 J 256 I 512 1 1024 I 2048 1 4096 

























• T 5 








































•5 1 






•5 1 


































oretical result (2.2), is noteworthy. The reason a definitely de- 
scribed source is necessary is that this time of reverberation could 
not be independent of the strength of the source, or the rate at 
which sound energy is emitted. It would really depend upon the 
sound intensity existing in the room at the instant of the dis- 
continuance of the emission. A standard initial intensity must 
therefore be adopted and these two equations were obtained for 
cases where the initial intensity is one million times that just 
audible. (The nature of the ear is such that the intensity which 
is a million times that just audible does not appear very loud.) 
In considering the phenomenon of reverberation it should be 
noticed that the intensity of sound builds up to its maximum value 
in the reverse manner to its decay. The intensity increases until 
the rate at which it is absorbed equals the rate of emission. 

2.6. Modern Absorbing Materials. — The significance of the 
researches of Professor W. C. Sabine in 1895 and of subsequent 
contributions from him and others were but slowly appreciated 
by architects and manufacturers. Prejudice was apparent and 
many insisted that the excellence of auditorium acoustics was 
really dependent upon the shape of the room. Given the proper 
ratios of the dimensions the results were claimed to be the same. 
Slowly and inevitably scientific facts, particularly the relation 
between volume and absorbing power, spread and today active 
interest in the subject is growing rapidly. Even Table II, or 
the list from which it is taken, quite inadequately represents the 
acoustic materials now available. The United States Bureau of 
Standards is actively engaged in the measurement of absorption 
coefficients and is making these values as well as general infor- 
mation on the subject available to the public* 

2.7. Absorption Coefficients. — Numerous absorption coeffi- 
cients have been measured. A partial f list is shown in Table II. 

* See circular of the Bureau of Standards No. 380, Jan. 4, 1930. Texts are 
Acoustics of Buildings by Watson, Architectural Acoustics by V. O. Knudsen, 
both John Wiley and Sons, 1930 and 1932 respectively, and Acoustics of Build- 
ings by Davis and Kaye, Ball, London, 1927. 

f These are taken from a list compiled by Dr. P. E. Sabine of the Riberbank 
Laboratories and published in Acoustics by Stewart and Lindsay, D. Van Nostrand. 




Here the variation of the coefficients with frequency and with 
arrangement of material are indicated. 

Formulas (2.1) and (2.2) assume that in determining V and 
a y the meter shall be used as the unit of length. Hence the com- 
puter must measure the volume in cubic meters and the areas of 
each kind of exposed surface in square meters. Each area must 
be multiplied by the corresponding coefficient of absorption and 
all these products added. If the foot is used as the unit, then 
the constant in (2.1) becomes .050, but the coefficients remain 
the same. An illustration in an actual case follows: 


Area in 
sq. ft. 


Abs. 1 

Wood sheathing, including all wood surfaces . . . 
Plaster on lath 

1 108 







137 i 


Plaster on tile 



Air vents •. 

Opera chairs, 922, absorbing power each, esti- 


Volume, 165, 200 cu. ft., therefore 

o.oco X i6c,2oo r 

t = — — = 6.24 

1324. 1 


The time of reverberation in this auditorium was measured and 
found to be 6.26 seconds. 

2.8. Absorption Coefficients and Frequency. — If the absorp- 
tion occurs in the air pores or channels in the wall of a room, 
then the absorption coefficient should change with frequency, 
and the influence of viscosity can be shown to be greatest in 
waves of short wave-length.f 

* These values were taken from measurements by Professor W. C. Sabine and 
were the only ones in existence at the time the computations were made. 
\ Rayleigh, Theory of Sound, Vol. II, Chapter XIX. 


In Table II the variation of the absorption coefficient is not 
always that of an increase with frequency. One concludes that 
the effect of viscosity in the pores is not the only important 
effect. In addition there may be viscosity in the material itself. 
At any rate, the absorption coefficient can be determined only by 

That painting a surface affects its quality is to be anticipated. 
The following * absorption Table III illustrates the effect of paint 
and of moisture. 

Table III 


Absorption coefficients 




























The column marked 1 contains values originally obtained 
by Dr. W. C. Sabine for an unpainted 18" wall of hard brick 
set in mortar; column two is for a surface of gypsum plaster 
with a so-called "putty finish" taken about three months after 
placing on an 18" brick wall and column three is for the same 
surface a year afterward. The data of the last two columns were 
obtained by Dr. P. E. Sabine. The change in the coefficient 
with time is doubtless caused by the evaporation of moisture. 

It is an interesting fact that a slight film of water or of any 
substance of great density as compared with air, will prevent the 
transmission of sounds. This is because the sound striking such 
a surface experiences practically total reflection. Thus it occurs 
that a thin film of water or vaseline will more effectually prevent 
the transmission of sound than a surprising thickness of highly 
absorbing hair felt. An interesting experiment may be performed 
by the use of cotton in the ears. The great difference produced 

* P. E. Sabine, Physical Review, 16, p. 514, 1920. 


by the additional use of a thin film of vaseline at the opening 
can then be observed. 

2.9. Other Effects in an Auditorium. — In the foregoing there 
has been discussed only reverberation. There are several other 
effects of importance, for example, resonance and the variation 
of intensity in various portions of the room. These will be men- 
tioned in later paragraphs. 


1. What is your reason for the statement on p. 24 that, inasmuch 
as there is no motion perpendicular to the plane, a wall can be sub- 
stituted therefor? 

2. In the discussion of echo, it was stated that the "deviation from 
a plane must not be large." Justify this statement in your own 

3. If the reflecting plane were corrugated, but still made up of 
small plane surfaces, could you find an image by treating each surface 

4. In a given channel, where would the velocity of the air particles 
be the greatest, and where the least? 

5. What is the relation between "reverberation" and "echo?" 

6. Show, from a consideration of equation (2.1), that the time of 
reverberation must be less in an auditorium without sidewalls. 

7. Why would the desirable time of reverberation in an auditorium 
depend upon the rapidity of speech? 

8. Give one reason for the opinion that a fog does not have the 
same effect as a film of water in preventing transmission. 

9. What objection would there be to a room in which all walls 
and objects are without absorption? 

10. If a room has too much reverberation, what advice should be 
given to a speaker? 

11. Show by assuming the walls large, that there is a multitude of 
images outside the walls of the room. 

12. A piece of cheese cloth is known to stop the wind in a marked 
manner. Why is it not effective in stopping the passage of sound? 

13. What phenomena in architectural acoustics have you observed 
that you can explain and that you cannot explain? 


3.1. Nature of Interference. — In Chapter I we discussed the 
nature of sound waves of displacement, pressure and velocity. 
The easiest manner in which to conceive of interference is by 
regarding each wave as something that is propagated and that 
produces at every point in its path its own displacement, pressure 
and velocity. Its effect must be independent of every other 
wave. If two sound waves cross, then the air particles at the 
crossing must have values of displacement, pressure and velocity 
that are resultants, or combinations of the two waves. Thus the 
two displacements must be added, proper regard being given to 
their directions as well as to their magnitudes. If two waves of 
the same frequency are travelling in the same direction and if at 
a point their displacements reach positive maxima at the same 
instant the two vibrations are in the same phase. If one reaches 
its positive maximum displacement and the other its negative 
maximum displacement at the same instant, the waves are said 
to be opposite in phase. At the point where two displacements 
are equal and in the same direction, the resultant is twice either 
displacement. Where the two are equal and in opposite direc- 
tions, the resultant is zero. In both cases the phenomenon is 
called "interference," for in both cases there is a combined effect 
which interferes with the occurrence of either displacement. 

As an illustration of interference assume two tuning forks 
having almost the same frequency, one 255 and the other 256 
vibrations per second. If the two are now held near the ear, the 
sound swells and diminishes to zero once each second, giving the 
phenomenon of "beats." This is explained in accord with the 
preceding paragraph, for if one has 255 vibrations in the same 
time as the other has 256 vibrations, then the latter gains one 
vibration in one second. Assume that at the beginning of a 



certain second the vibrations of the two forks are in the same 
phase. Then one-half a second later, they will be opposite in 
phase. At the end of the first second they will be again in the 
same phase. In the next second the same cycle will occur. Thus 
during each second there occurs complete agreement in phase and 
exact opposition in phase. In the former case displacements add 
and in the latter they subtract. If these displacements are equal 
in magnitude we have in the former case four * times the intensity 
of the sound from one fork and in the latter case no intensity at 
all. This effect gives us the phenomena of "beats" to which 
reference has been made. 

3.2. Huyghens* Principle. — There are many interesting phe- 
nomena which are explicable only when one examines the inter- 
ference of the sound waves. If sound travels with equal velocity 
in all directions from a simple source of sound, we say that a 
spherical wave results. But a sphere drawn about this source 
has a significant uniqueness. At every point on this sphere the 
vibration is simultaneously in the same phase, for since all points 
are equally distant from the origin, the displacements must be the 
same. To repeat, every point on this sphere is vibrating in the 
same phase. Such a surface is called a "wave front." Usually, 
in acoustics, the direction of propagation of a sound wave is 
perpendicular to the wave front; the exceptions of interest will 
be subsequently noted and the reason therefor given, but for the 
present purpose this direction of propagation will be assumed as 
correct. With this understanding, it is easy to designate a wave 
front and thus to determine the direction in which the wave is 

What is termed Huyghens* principle is that each point on any 
wave front can be assumed to be the source of a hemispherical 
wave and a future wave front be thereby determined. This is 
illustrated effectively in Fig. 1.4. This principle is capable of a 

* Technically, the intensity, frequency constant, is proportional to the square 
of the maximum displacement. See footnote to Section 2.3. This maximum value 
is called the "displacement amplitude" or frequently the "amplitude." 



rigorous proof in acoustics, but here the principle will be assumed. 
Consider the spherical wave a a in Fig. 3.1 and select a number of 
points along this cross-section of the wave front. With each 
point as a center draw a semi-circle, each 
semi-circle having the same radius, equal to 
the distance traversed by the waves in some 
definite time. It is our task to ascertain the 
new wave front. Inspection shows that at 
the instant considered, at no other surface 

same phase. For assume that the distance 

between a a and a! a! is "#" wave-lengths. 

Then draw any other curve you please from 

the upper point a' to the lower point a\ 

Remembering that only points an integral 

number of wave-lengths apart in the radial p IG - x 

direction can be said to be in the same phase, 

one sees that ^//points on this newly drawn curve cannot possibly 

be in the same phase. Hence we cannot regard it as a wave front. 

3.3. A " Beam " of Sound.* — Assume we have a vibrating 

area, ab, in a plane wall as in Fig. 3.2. Assume that it is large 

in comparison with a wave-length of the frequency actually used. 

According to the previous paragraph we may now construct the 

later wave front a'b'. It is readily seen that the hemispherical 

waves from all points along ab> travelling in the directions ac or 

bdy would not have a surface in common, such as a'b' . In other 

words, the hemispherical wavelets are not in agreement as to 

phase in the directions ac or bd. This introduces interference and 

if ab is as long as many wave-lengths, it can be seen that there is 

destructive interference in these two directions. For, consider 

the direction more nearly along the wall aA. At a point P the 

displacement produced by the element of area at a will be equal 

and opposite to that produced by the element of area one-half 

* This can be illustrated by means of a highly pitched whistle placed inside 
and at the closed end of a cylindrical tube. The open end may be considered 
approximately a wave front. 



wave-length further away from P and between a and b. If ab 
is comparatively large, as has been assumed, there will be as many 
elements producing a displacement of one phase as there are ele- 
A ments producing a displacement of op- 

• P posite phase. There is, then, approxi- 

^~'" mate annulment at P. What is true of 
^~""*" the waves in the direction aP is true of 

^d any direction other than aa! or bb\ 

Thus if ab is comparatively large, it will 
send out a "beam" of sound a'b\ similar 
to a beam of light sent out by a search- 
J> ', light. Of course, the edge of this beam 

! **^^ is not sharp, for ab is not infinitely large 

j ^ % \*/ compared to a wave-length. This ex- 

| planation of the condition which will 

• produce a beam of sound demands care- 

Fig. 3.2 fal study, for similar reasoning will sub- 

sequently occur. 
If ab is now reduced in size, it is readily seen that the resulting 
wave front becomes more and more like a hemisphere. When 
ab is a point source, the wave is hemispherical. 

As an application of the above reasoning, consider the direc- 
tive property of a megaphone. It is known that the surface 
containing the large opening is a wave front. If the opening is 
large as in a large megaphone, the intensity produced is distinctly 
greatest along the axis of the megaphone. There is never a 
noticeably sharp beam of sound, and this is true because the 
wave-lengths used are not sufficiently small. 

3.4. Acoustic Plane Reflector. — If reference is made to Figs. 
1.2, 1.3 and 1.4, a series of ripples reflected from the wall will be 
noticed. This section is a brief study of the influence of the size 
of an area upon the reflection of acoustic waves therefrom. Con- 
sider Fig. 2.1. is a source of sound in front of an infinite wall. 
Its reflected wave can be regarded as coming from 0'. If, how- 
ever, we are dealing with a reflector of ordinary size, or a part of 


the wall, the reflected wave cannot be the same as that coming 
from 0', in the absence of the reflector. It must be of less inten- 
sity because it is jf less area. According to Section 2.1 each area 
of the infinite wall will give a reflected wave which is just like 
the wave which would come from 0' through this same area in 
the absence of the wall. Consider this hypothetical wave from 
the image 0' in Fig. 3.3, passing through the circular area ab, 
which has replaced the small reflec- 
tor of the same size. This wave 
will not remain in the conical volume 
indicated by the dotted lines. Fig. 
1.2 has already illustrated an analo- 
gous action in the case of ripple 
waves. The section just preceding 
considers the divergence from a geo- 
metrical beam in the case of a plane 
wave front. Obviously this diver- 
gence will occur with a portion of a p IG , - 
spherical wave as well. Conse- 
quently the wave front in ab will spread outside of the cone, 
unless the distance across ab is long compared to a wave- 
length. Moreover, the smaller ab the more the waves will 
diverge from it as from a point. Certain conclusions are 
now evident. Not only will a small reflector reflect a small 
amount because of its size but a small reflector will scatter sound 
in all directions , thus further greatly reducing the sound reflected 
backward. This shows that, as the area of a small reflector is 
increased, there is a much more rapid increase in its effectiveness. 
Yet another influence should be mentioned. The reflected wave 
will scatter not only in all directions on the same side as the 
reflector, but also around the reflector itself. 

3.5. Acoustic Parabolic Mirror. — It is shown in optics that 

light from a source located at a certain point within a parabolic * 

mirror called a "focus," will be reflected from the mirror as a 

* The parabolic mirror has a concave shape similar to that of an automobile 
headlight. Its property of focussing is the important point and not its shape. 


parallel beam. This property of a parabolic mirror is utilized 
in searchlights, the arc being placed at this focal point. Also 
light from a distant source such as the sun will be brought to 
this focus. During the war, acoustic parabolic reflectors were 
tried as sound detectors, since the waves from a distant source 
are approximately "parallel" and should be reflected to a focus. 
Huge mirrors 12 feet in diameter were constructed. But it was 
found that the concentration of sound was very poor. From our 
preceding explanations it is evident that sound will be reflected 
as will light only if the reflector is of dimensions very large in 
comparison with the wave-length. Moreover, in light, the small- 
ness of the focus is caused by interference. The intensity is 
greatest where all the waves are in the same phase. The dis- 
tances from likeness of phase to opposition in phase is of the same 
order of magnitude as a wave-length as will later be shown when 
discussing two waves travelling in opposite directions. Thus it 
occurs that the focus of an acoustic mirror is not sharp and may 
be regarded as of approximately the same diameter as a wave- 
length. If we have a plane wave of frequency 100 d.v. (complete 
or double vibrations per second) striking a 12 foot mirror, the 
focus would have approximately the diameter of 1100 -f- 100, or 
n feet. This can scarcely be said to be concentration. Even 
with a note of high frequency, 1000 d.v., the diameter of the focus 
would be 1 foot. With a highly pitched whistle, a parabolic 
mirror and a sensitive flame, the concentration of sound may be 

Small parabolic mirrors are sometimes used behind public 
speakers. They produce a noticeable difference but are not very 

3.6. Interference in Auditoriums. — Our discussion of inter- 
ference of sound leads to an explanation of the well-known fact 
that the intensity of sound is not equal at all parts of an audi- 
torium. In an enclosed space there are innumerable reflections 
and at any point the resultant intensity may be regarded as pro- 
duced by many sources (images). But, if sound waves may inter- 



fere so that displacements rather than sound intensities must be 
added, the intensity of sound will not be the same throughout. 
Dr. W. C. Sabine was the first to explore the variation of inten- 

Fig. 3.4 

sity throughout an auditorium. He has represented the varia- 
tion in intensity in the manner that elevation of land is shown in 
topographical maps. Fig. 3.4 is taken from his work. In a pre- 


vious chapter it was shown that the time of reverberation can be 
computed in advance with considerable accuracy. Acousticians 
have not yet learned a simple way to ascertain the presence of 
" bad " spots or areas of small intensities. Moreover, the location 
of such spots will vary with the frequency. It is possible, how- 
ever, that a considerable area in an auditorium may be, in general, 
a poor place for an auditor. 

3.7. Selective Property of Reflectors. — There is a curious 
property of reflectors to which this section will be devoted. 
Assume we have a vibrating disc, Fig. 3.5, and a point of obser- 
vation at 0. For the sake of simplicity the emission from only 
the right side of the disc will be considered. The effect at will 
be due to the combined effects of all portions of the circular disc 
AB. But the sound from A will reach the point later than the 
sound from C. Then the phase of the former will not be exactly the 
same as of the latter. If we increase the size of the disc, the 
intensity at will increase unless the added area produces a dis- 
placement at that nullifies a portion of the composite displace- 
ment already produced. Just at what size this will occur can be 
seen by the following discussion. 

Suppose the circular area shown in cross section by ACB in 

Fig. 3.5 be divided up into con- 
centric circles, in such a manner 
that the second circle adds an 
area that is exactly equal to the 
first one drawn about C, the 
third also adds an area equal to 
Fig. 2-5 tne ^ rst an d so forth. Then 

each successive addition may be 
thought of as contributing an equal amount to the resulting am- 
plitude at 0. The whole area is vibrating in the same phase, 
but the above areas are at constantly increasing distances from 
0. Hence, the above successive contributions to a resultant 
amplitude must gradually decrease in phase. It can be shown 
that every additional area adds to the resulting amplitude until 


Fig. 3.6 

the phase reached is opposite to that of the wave from C, when a 
decrease in the resultant begins. That the reader may have a 
better picture of this effect, the following discussion is given. 
If one adds two equal displacements as in Fig. 3.6, R, the diag- 
onal, is the resultant. The resultant 
may be regarded as obtained by ad- 
ding the two arrows, r% and r 2 , end 
to end and connecting the terminus 
with the origin. This process may 
be continued with a third arrow. 
Suppose one adds a series of dis- 
placements as in Fig. 3.7, each equal in magnitude but differing 
in direction. The resultant is found by attaching the arrows end 
to end and drawing R ly R 2y etc. It is noticed that the last dis- 
placement that makes a con- 
tribution to the length of /?, 
ef, is opposite * to the initial 
displacement Oa. This method 
of adding displacements may 
be justified by a consideration 
of the nature of the variation 
of a displacement having the 
simple harmonic variation vis- 
ualized in Section 1.7. There 
such a variationgivas described 
as corresponding to that of 
the shadow of the displace- 
ment of a ball on a screen, 
the ball itself moving in a 
circle in a plane containing the direction of the light. This 
simple harmonic variation of displacement is the one treated in 
this section and throughout the text. For, as stated in Section 
1.7, a complex sound is made up of such units, and the effects 
produced by each unit may be added to determine the result 

* According to Fig. 3.7, the last one to make a contribution to R is de> but if 
the areas are made smaller and smaller, de more and more nearly approaches the 
condition of opposition in phase. 

Fig. 3.7 


occurring with the complex sound. Consequently the treatment 
of reflection for a displacement having one frequency may be 
regarded as really of general application. Retaining the experi- 
ment with the ball in mind, consider the addition of two displace- 
ments differing in phase but represented on the same wheel. 
Assume the second displacement to have the same amplitude 
also indicated by the radius of the wheel. If the two displace- 
ments are opposite in phase, their corresponding radii are drawn 
in opposite directions. They have an angle between them of 
180 . (It is customary to use such an angle as expressing nu- 
merically the difference of phase of the two displacements. Thus 
a difference in phase may be designated as 30 , 40 or 210 , as 
the case may be.) These two displacements, opposite in phase 
and equal in amplitude, will give zero if added. The addition 
can be made graphically by drawing the two radii in opposite 
directions. Assume that the difference in phase is 90 instead 
of 180 . The two balls are placed at the rim with an angle of 
90 between the radii. The sum of the displacements at any 
instant can be found by adding the two values as found on the 
shadow on the screen. The resulting displacement evidently has 
a larger amplitude than the radius of the wheel. It can be shown 
by geometry that there is a radial line, drawn from the center of 
the wheel, which will have the value of this amplitude. More- 
over, its shadow on the screen as the wheel rotates will always 
give the resulting displacement produced by the two components 
described. This radial line can be proved to be the diagonal of 
the parallelogram having the two indicated radii as sides. This, 
then, is the manner of securing a resulting amplitude, when two 
displacements are in the same straight line. The amplitudes are 
represented by arrows, the difference of phase by the included 
angle, and the resultant amplitude by the diagonal. This is 
precisely what was done in Fig. 3.6 and by an extension in Fig. 
3.7. R is the resulting amplitude. 

In Fig. 2-S tne displacements at from the successive areas 
do not differ appreciably in amplitude or in direction if CO is 
long compared with CA. But they do differ in phase. If the 


angle between the direction of two arrows in Fig. 3.6 represents 
the phase difference and the equal lengths of the arrows, the am- 
plitude, then R is the correct resultant amplitude. Figure 3.7 
then states that, beginning at the center area of the vibrating disc in 
Fig. 2- Si the displacements at produced by the successive areas 
conspire to increase the resulting amplitude at that point until 
opposition in phase is reached (180 in Fig. 3.7). Additional 
areas decrease this amplitude at 0. A nullification will begin to 
occur when the difference in the paths OC and OA will cause the 
displacements from C and A to be opposite in phase. This 
critical difference in the paths is one-half a wave-length. When 
AC is increased beyond this critical distance the displacement at 
will diminish. A more extended study shows that the resulting 
intensity at 0, if the radius of the disc is gradually increased, will 
diminish from the above described maximum to a minimum, 
which is always greater than zero, will again increase and, in fact, 
will have a series of maxima and minima of less and less promi- 
nence, the intensity always remaining noticeably less than the 
maximum intensity occurring with the radius such that AO-CO 
is one-half a wave-length. This means that the action of the 
diaphragm is selective. For a given frequency there is a certain 
diameter which will give the maximum effect at 0. This is speci- 
fied by the condition that AO-CO is one-half wave-length. 

If this interesting effect occurs with a vibrating diaphragm, 
it will of course occur with a reflecting surface as well, for the 
latter may be considered as sending out a wave. The following 
unpublished experiment was performed by the writer out of doors. 

A circular screen S in Fig. 3.8 was placed about a cylindrical 
tube, R, from whose base a rubber tubing passed to the appara- 
tus for measuring intensity. The tube R had the length which 
caused it to resonate with the frequency of the source placed at 
0. The observer compared the intensities in R and hence at P 
when the size of the screen was altered. Three annular rings, 
adding successive portions of the enlarged area of S, were con- 
structed. The dimensions were adjusted so that the sum of the 
distances from P to the edge of the screen and from the edge to 


O was as follows: (i) Without any of the rings this total distance 
was less than one-half wave-length greater than OP; (2) with one 
annular ring it was exactly one-half wave-length greater; and (3) 
with the second and third annular rings it was more than one- 
half wave-length greater than OP. The distance OS + SP is 


• P 90 

Fig. 3.8 

regarded as a path of the sound wave, for the wave which is 
reflected at $ travels therefrom in all directions. As would be 
expected from the preceding discussion, the greatest intensity 
observed in R was with one annular ring or with the difference in 
total distance one-half wave-length. 

This selective property, namely, a maximum reflection for a 
given frequency, is, at present, more of a curiosity than a utility. 
Nevertheless, it sometimes occurs in echoes. 

3.8. The Pinnae as Reflectors. — The reflecting power of the 
pinnae, or the auricles of the ear, is not of grave importance. As 
already pointed out both the incident sound and the reflected 
sound bend around small obstacles. In fact, the effect of an 
obstacle does not begin to be very marked until its size is com- 
parable to a wave-length. Thus it is seldom that one listens to 
sounds of high enough frequency to permit the pinnae to be very 
effective. If one places a source of sound of 10,000 d.v. first in 
front and then behind the head, a marked difference in intensity 
can be noted. The wave-length is, in this case, less than the 


diameter of the pinnae. Similar remarks may be made in regard 
to the effect of cupping the hand at the ear. The reader may 
readily try experiments with the tick of a watch, which contains 
tones of high frequencies, and with ordinary sounds such as 

3.9. Acoustic Horns as Reflectors. — Contrary to the com- 
mon view, a megaphone, used either to transmit or to receive, 
owes its advantage in increasing the sound intensity not to a 
reflection effect, such as would occur if the wall was silvered and 
light was used instead of sound, but rather to a resonating prop- 
erty which will be discussed in a later section. In fact, the dis- 
cussion in this chapter shows that we cannot use the laws of light 
reflection unless all surfaces are very large in comparison with a 
wave-length. The directivity of a megaphone depends upon the 
area of the large opening, for this area is a wave front. Whether 
or not a good beam of sound is obtained may be determined by a 
discussion similar to that in Section 3.3. 


1. If two waves are travelling in the same direction, under what 
condition are the two displacements at a given point in the same 
direction at all times? 

2. If two waves are travelling in the same direction, under what 
condition can the displacement at one point be in phase, when simul- 
taneously, at another point, they are opposite in phase? 

3. In the case of a conical megaphone the text does not state 
whether the wave front at the opening is a plane or a section of a 
sphere. What would be the difference in effect if such two surfaces 
did not differ in position by more than one inch, for example? 

4. In Fig. 2.1, assume the wall were made up of many small planes 
set in somewhat random orientations. Under what circumstances 
would this not materially affect the location or sharpness of the image, 
07 Could this construction, if made of polished metal surfaces of 
an inch in diameter, prove a satisfactory mirror for light? 

5. With a small reflector why does the amount reflected depend 
upon the wave-length? 

6. When a train or automobile rushes by small objects such as 
tree trunks, telephone posts, bridge structures, are the sounds reflected 
to the passenger similar to those one would hear were he standing 
nearby on the ground, or what difference is noticed and why? 


7. Can a blind man by hearing tell anything concerning the nature 
of the objects along the path upon which he is walking and why? 

8. Assuming that clearness of enunciation depends upon the higher 
frequencies in the complex sounds, which will a small reflector improve 
the more, loudness or clearness of speech? 

9. Why should the bad spots in an auditorium vary in position 
with the frequency? 

10. One listening to a speaker talking through a megaphone will 
notice that there is a difference in the quality of voice caused by 
changing the direction of the megaphone relative to an auditor. State 
the difference that occurs and why. 

11. If one talks through a megaphone rectangular in cross-section, 
in which directions from the axis of the megaphone will the sound 
spread the most easily? 

12. Experiment with cupping the hands at the ears in order to 
improve hearing and report the nature of the sounds when distinct 
improvement was and was not made. 

13. Is the presence of the head any acoustic advantage in increas- 
ing the intensity at the ear? 

14. What does the experiment illustrated by Fig. 3.8 show as to 
whether or not sound is reflected like light? 


4.1. Variations of Velocity in the Atmosphere. — Equation 
(1.1) shows the dependence of the velocity upon pressure, density 
and the ratio of the specific heats of a gas. In equation (1.3), 
however, we have an expression for the velocity in which neither 
the pressure nor the density of a gas occurs. In the atmosphere, 
which will here be assumed to have everywhere the same com- 
position, there are changes in pressure, density and temperature. 
But equation (1.3) states that in discussing the velocity of sound 
in such a non-homogeneous medium at rest we need to consider 
only those variations in velocity of sound that are caused by 
variations in temperature. If there is a wind having a direction 
parallel to the earth's surface, for example, obviously the velocity 
of the sound wave is greater, relative to the earth, when the prop- 
agation of the sound is in the direction of the wind and less when 
the sound travels to the windward. In considering the propaga- 
tion of sound in the atmosphere we have need, therefore, to take 
careful account of both the temperature and of the wind. 

4.2. Effect of Temperature. — Equation (1.3) shows that the 
higher the temperature of a gas, the greater the velocity of sound 
therein. The influence of temperature may be studied by assum- 
ing the case of a sound wave passing from one stratum at a tem- 
perature / to another stratum at temperature /', /' > t, giving 
velocities v' and v. This is done in Fig. 4.1. Here the horizontal 
line represents the plane separating the two media. The initial 
plane wave front is AB and the final wave front after what is 
termed "refraction" is A'B\ The change of direction of propa- 
gation occasioned by a change in the nature of the medium which 



affects the velocity of sound is called "refraction." * This term 
is also used in cases where there is a change in velocity even if 
the direction of propagation remains unaltered. Such would be 
the case if BB' were perpendicular to the boundary in Fig. 4.1. 
The position at A'B' is obtained in the following manner. Since 
the sound will travel the distance A A' in the same time that it 

will travel the distance BB' y the wave front must contain the 
point B' and a point on the hemisphere about A with a radius 
A A \ If it is now assumed that the resulting wave front is plane, 
which a more extended discussion would justify, then A'B' is the 
refracted wave front. It is thus seen that the direction of propa- 
gation of the wave in the second medium, A A \ is perpendicular 
to the wave front, A'B'. 

This effect of temperature produces an interesting result which 
is experienced usually in the early morning hours. If the night 
has been clear, the earth has been radiating heat rapidly and, in 
the absence of wind, the atmospheric layer near the ground may 
become cooler than that above. This is just the reverse of the 
usual daytime condition when the heat from the earth causes the 
lowest layer to have the highest temperature. When the layer 
adjacent to the earth has been cooled until its temperature is 

* There will also be a reflected wave in the first medium, but it is not discussed 
at this point. The case of reflection (in gases) with perpendicular incidence is 
discussed in Section 5.4. In general all the energy will not pass from the first to 
the second medium. 


lower than the stratum above, the temperature in this lowest 
region will increase gradually with elevation until a level is 
reached where the temperature begins to decrease. At further 
elevations the decrease continues indefinitely, so far as our pres- 
ent acoustic interest is concerned. Within the layer of tempera- 
ture increasing with elevation, we have sound refracted in a 
manner similar to Fig. 4.1. In the discussion of this figure it 
will be assumed that the transition in temperature occurs sud- 
denly. It can be proved that in the case of a gradual temperature 
variation we are justified in using the above conclusion qualita- 
tively. It is therefore seen that the effect of this type of refrac- 
tion is to bend the sound towards the earth. If the sound is 
propagated in a direction more nearly horizontal than shown in 
Fig. 4.1, it may reach a horizontal direction and then be refracted 
downward before reaching the upper limit of the stratum of tem- 
perature increasing with elevation. If so, the sound will be 
retained within the stratum very much as if transmitted between 
two parallel walls, one of them the earth. This accounts for the 
great distances at which sounds may be heard in the early morn- 
ing hours. In a similar manner it can easily be shown that when 
the stratum is in its usual daytime condition the influence of 
temperature is to decrease the range. 

Attention should be called to the fact that, in the foregoing 
paragraph, there has been described a case of what is essentially 
"total reflection" caused by the phenomenon of refraction. But 
one must not accept too literally the word "total." There will 
be sound diffused to some extent out through the layer, irre- 
spective of the direction of propagation of the transmitted sound. 
The possibilities in Fig. 4.1 may be examined more closely. Sup- 
pose the temperature of the upper medium be gradually increased. 
At each increase A A 1 becomes longer while AB' and BB' will 
remain unchanged. Thus A'B' will change its position and the 
direction of propagation A A* will become more nearly horizontal. 
As the temperature is further increased, causing v' to become 
greater, AA' will approach the length AB' . When AA' equals 
the length AB' the direction of A A' becomes horizontal. This 


condition means that the wave no longer enters the second me- 
dium but skims the surface between the two. If the temperature 
is now further increased the sound will be reflected at the bound- 
ary. The reader must not think such a reflection is impossible 
though we do not commonly have such large temperature differ- 
ences as would be required for the angle of incidence used in 
Fig. 4.1. Instead of requiring the temperature of the upper 
medium to be raised, one may change the angle at which the 
direction of propagation of the wave meets the boundary. As 
this direction becomes more nearly parallel to the boundary the 
refracted wave also approaches coincidence with the boundary 
even more rapidly, then skims the surface and finally is totally 
reflected. So it appears that even a small difference in velocity 
in the two media may, if the grazing angle of approach is small 
enough, produce total reflection. This is further discussed in 
Section 5.1 1. 

4.3. Effect of the Wind. — Assume that in Fig. 4.2 the line 
00' is the cross-section of a plane which separates two regions of 
air, in the upper of which the medium is moving with a velocity 

relative to the earth of Ui and in the lower with a velocity of #1, 
u 2 being greater than u\. Assume further that there is a plane 
sound wave represented by the cross-section of the wave front, 
AB 3 which is moving from the lower into the upper region. If 
the air everywhere were stationary, the wave front would move 
from AB to A'B' in a certain time, say /. But simultaneously 


with this propagation each portion of the wave front will be 
carried forward by the motion of the medium in which it is lo- 
cated. At the expiration of the time, t, when the wave should 
have reached A'B\ it will actually be at A"B"> for the distances 
A' A" = u 2 t and B'B" = u x t are the distances which the respec- 
tive media have transported their sound waves in the same time. 
Because of the fact that u 2 is not equal to u u A"B" is not parallel 
to AB and we discover that the wave front has changed its angle 
with 00'. It is further easy to see that the direction of propaga- 
tion from 00' is not perpendicular to A"B", or the wave front, 
but is in the direction AA". We can conclude that in a gaseous 
medium having convection currents, the direction of sound prop- 
agation is not, in general, perpendicular to the wave front. 

But suppose an observer were moving with a velocity u 2 , that 
is, with the upper medium. He would notice nothing unusual in 
the upper medium. The direction of propagation relative to him 
could be drawn normal to the wave front. Thus we see that this 
peculiarity, the movement of the wave front not perpendicular 
to itself, may be caused by the motion of the observer relative 
to the medium, and thus not be strictly an effect of the medium 
itself. But if the velocity of the medium varies from point to 
point in any direction, this peculiar movement of the wave front 
may exist and may be correctly attributable to the medium itself. 

It is to be understood that the abrupt change of wind velocity 
with elevation, as here supposed, cannot really occur, and that 
refraction actually extends over a considerable depth in which 
the wind velocity varies gradually. 

A practical point naturally arises as to the direction of propa- 
gation of sound as judged by an observer who judges wholly by 
hearing. As pointed out in a later chapter, the sound appears 
to come from a direction perpendicular to the wave front. As will 
appear, this is because equality of phase of vibration at the ears 
is the deciding factor. 

4.4. Speaking in the Wind. — It is a simple step to the dis- 
cussion of the influence of the wind upon the propagation of sound 


horizontally, and this is the case of out-of-doors speaking. In 
the diagram of Fig. 4.3 assume the source of sound to be at S 
and located in a wind having a velocity to the right that increases 
with elevation above the ground, and assume the wave which 

has proceeded from it 
to be AB. Proceeding 
as before, the lines AA' t 
c\ c' \c" CC and DD' are drawn 

as the distances that the 
sound wave would travel 
in the time / if the 
. motion of the air were 

p IG « nil. But the- medium 

has moved in this time, 
/, a distance A' A" at the upper, CC" at the center and zero or 
a very small amount at the lower point. The considerable 
variation in wind velocity that is here represented really exag- 
gerates the actual case, but, 
inasmuch as the wind veloc- 
ity does increase with the 
vertical height, the assump- 
tion made for the variation 
will give us results that are 
qualitatively correct. It is 
seen that the wave has 
reached A"~D' in the time /, 
and is now a distorted spher- 
ical wave. Moreover, the direction of propagation from A is not 
AA' but A A". In other words, the wave is bent or refracted 
toward the earth. The result of such refraction is the reduction 
of the spreading of the wave upward from the horizontal. The 
effect, therefore, is that the speaker's voice, or any other sound 
from S, is carried apparently along the earth. It is said his voice 
carries better in the direction of the wind. 

In Fig. 4.4 is illustrated the effect when the sound travels 
against the wind. The reader can follow reasoning similar to the 

Fig. 4. 


above without the necessity of its repetition here. The conclu- 
sion is that the sound wave is refracted upward, thus making it 
more difficult to be heard when speaking against the wind. 

The wind may cause effects that at first thought seem curious. 
For example, two persons may be attempting to communicate 
in a high wind between shore and boat. They find that while 
the individual leeward can hear and understand, the other to 
windward can scarcely hear any sound whatever. In fact, sound 
will pass "more readily" in one direction than in the opposite. 
But if the observer to the leeward is elevated he will be able to 
make the observer to the windward hear much better, for now, 
so to speak, the sound may be refracted but will yet succeed in 
reaching the hearer. Such would be the case with sound travel- 
ling from the source in a somewhat downward direction. Ele- 
vated church bells can thus be heard more distinctly at points 
windward than if the bells are near the ground. The refraction 
toward the ground of the sound travelling leeward is not so seri- 
ous a matter, for, assuming no obstacles, reflection will occur 
and the sound does not escape as in the case where the sound 
travels to windward upward. Of course, the presence of obstacles 
on the ground is an additional justification for the elevation of 
bells and whistles. 

The previous discussion in regard to the scattering or diffusion 
of sound showed that in acoustics we rarely deal with a "beam" 
of sound. Therefore it is incorrect to suppose that sound trans- 
mitted to windward entirely leaves the ground. Likewise, it is 
impossible for all the sound transmitted to leeward to be refracted 
toward the ground. The effects as described are not complete. 

4.5. Silence Areas. — In view of the influence of both wind 
and temperature upon sound refraction it is not surprising that 
conditions may obtain wherein the noise of an explosion may be 
distinctly heard in a distant place, and yet not heard at all at a 
nearer position. This phenomenon has been frequently observed. 
As an illustration on October 28, 1922, 2,000 pounds of explosives 
were fired in Holland. The noise was heard within a radius of 


from twenty to seventy kilometers, the difference in different 
directions being caused by the wind. No sound was heard be- 
tween seventy and two hundred kilometers^. But in a zone dis- 
tant more than two hundred kilometers the sound of the explosion 
was again audible, in fact, up to a distance of nine hundred 

4.6. Refraction and Scattering of Airplane Noises. — In lis- 
tening to airplanes in flight one observes several acoustic phe- 
nomena. The one most quickly noticed is that the sound from 
the airplane is of an uneven character. During experiments * 
in connection with airplane detection and location the observers 
noticed also that, with "poor listening" atmospheric conditions, 
the sound from the airplane at the greatest hearing distance was 
limited to the lowest frequencies in the emitted complex sound. 
These frequencies were for these particular airplanes approxi- 
mately 90, 180, 270, etc., and the most prominent component 
was the one of lowest pitch. The sound from the same airplane 
heard at the greatest possible distance under excellent night con- 
ditions was distinctly different. The lowest frequencies just 
named were not noticeable and the sharp crackling sounds of the 
engine explosions with prominent components, probably of the 
order of 1,000, were most distinctly in evidence. The difference 
in the character of the sound in the two cases may be described 
as the cutting off of the higher frequencies in the former and of 
the lower frequencies in the latter. That there may be a more 
rapid decay of intensity of the higher frequencies is readily under- 
stood by a consideration of differences in wave-length. For the 
irregularities in the planity of the strata, for example, would be 
more effective in scattering, by reflection and refraction, the fre- 
quencies having the shortest wave-lengths. The reader can dem- 
onstrate this if he will make a drawing of a non-planar stratum 
and consider the refraction of waves in detail. In general, the 
irregularities of the atmosphere would effect more the shorter 

* See Stewart, Phys. Rev., N.S., Vol. 14, No. 4, p. 376, 191 9. 


The apparently better transmission for the higher frequencies 
in the second instance is not to be explained by any influence 
of the medium but rather by the characteristics of audition. 
The sense of loudness for the different frequencies is not the same, 
whether the intensities * are measured in mechanical units or in 
terms of the least audible intensity. It is the latter unit of meas- 
urement that is of interest in the present case, for the nature of 
the sound heard at a great distance from the source depends upon 
audibility. A further explanation will be given in a discussion of 

The other experimental fact worthy of record is the rapidity 
with which the intensity falls off with distance in the atmosphere. 
If sound energy given off by the engine occurs at a constant rate, 
and if this energy spreads out in a spherical wave, then the amount 
of energy per unit volume or the intensity would vary inversely 
as the area of the expanding sphere. This means that the inten- 
sity varies inversely as the square of the distance from the source. 
In the experiments to which reference is here made, the observer 
used an instrument that amplified the sound intensity one hun- 
dred times. If the intensity of sound varied inversely as the 
square of the distance from the source, then the observer should 
hear an airplane at ten times the distance one could hear it with 
the unaided ear. But on fair days, cumulus clouds forming, 
airplanes at an elevation of one thousand yards could be heard 
only twice as far with the instrument as with the unaided ear. 
On days when the atmosphere was obviously more irregular, the 
decay of intensity was much more rapid. Under good night 
observing conditions with the airplane at an elevation of two 
thousand yards the maximum distance of hearing was increased 
to three times the distance possible with the unaided ear. 

It thus appears that even when the atmosphere is favorable 
to sound transmission there is sufficient irregularity to cause a 
surprisingly rapid decay of sound from an elevated source. 

4.7. Diffraction. — The word diffraction is used when a 
change of direction of propagation of sound is occasioned, not 
* By "intensity" is meant the energy per unit volume. 


by a difference in the medium itself but by the introduction of 
obstacles or reflecting surfaces, causing the sound to bend or to 
diffract around such objects. Diffraction has, as a matter of fact, 
previously been mentioned in Sections 1.3 (with ripples) and 3.4, 
but without being so designated. Sound will pass around the 
corner of a building or over a partition or out of a window. 
Everyone knows that the hearer need not see the source of the 
sound. Obviously, diffraction is a very common and also an 
important phenomenon. Without it we would be put to great 
inconvenience in the conduct of our daily affairs. Diffraction of 
sound must have played a significant role in self protection in the 
evolution of the race. 

From our earlier discussion in 3.4 in regard to the dependence 
of the efficiency of reflectors upon the frequency of the sound 
concerned, it can readily be judged that the longer the wave- 
length the less the reflection and the greater the diffraction pro- 
duced by an obstacle. It will be recognized also that our dis- 
cussion of the reverberation in auditoriums assumed the free 
penetration of all frequencies to every corner and recess in the 
room. Fortunately the diffraction phenomenon is sufficiently 
pronounced to make this assumption approximately true. 

4.8. Diffraction about the Head of a Speaker. — The diffrac- 
tion about the head is an important consideration in two cases, 
speaking or singing and audition. In the former the source is 
at the head and in the latter it is removed at a distance. Results 
concerning diffraction about the head will now be discussed. 
The theoretical investigation, first begun by Lord Rayleigh and 
later continued by others,* assumes the problem to be the inves- 
tigation of the diffraction of sound about a rigid sphere. This 
shape is necessary to make the mathematical solution possible. 
The term "rigid" means simply that the sphere is not set in vi- 
bration by the incident sound and hence that there is no absorp- 
tion at any point on its surface. Let the circle in Fig. 4.5 repre- 

* Stewart, Physical Review, 33, 191 1, 467-479. 
Hartley and Fry, Bell System Technical Journal, 1, 33, 1922. 


sent the cross-section of the rigid sphere, the point A, the source 
of sound on the sphere, and P and P' positions of the observer, 
the former on the same radial line as A and the latter on a radial 
line making an angle 6 with OP. The inquiry is now made as 
to the intensity of sound from A observed at any point P'. A 
glance at the problem shows that the solution is not easily ob- 
tained by physical reasoning or by utilizing any of the facts 
previously studied herein. The sound will spread out and will 

Fig. 4.5 

pass in every direction from A around the sphere. Moreover the 
bending or diffraction will not cease when the sound has gone 
half way around the sphere. There is no reason why the wave 
should not continue to diffract around the sphere to the front. 
In fact, the effect at any point may be considered to be caused 
not only by the sound being diffracted around and close to the 
sphere repeatedly, but also by the entire wave front passing the 
sphere. The phenomenon seems hopelessly complicated. But 
the mathematical solution is relatively simple in principle. The 
method briefly stated is not without general interest. The equa- 
tions describing the acoustic nature of the medium are set up. 
Then it is assumed that the correct solution must "satisfy" these 
equations and also the condition of no radial motion at all points 



of the sphere except at the source, A. The solution can then be 

In Fig. 4.6 are shown curves representing the results computed 
from the theory and expressed in the form of the ratio of intensity 
at any point P' in Fig. 4.5 to that at the fixed point, P, the dis- 
tance OP being equal to OP' and the angle 6 lying between them. 
The sphere is assumed 60 cm. in circumference. Let us consider 
curve 2. Here P' is 19.1 meters from the center of the sphere. 
As P' moves about the sphere in a plane the intensity at P* 




<8 .60 


Curve 1 As 120 cm. rVoo 

2 M20 cm. . r « 1910 cm. 

3 As 120 cm. r» 477 cm. 

4 As 120 cm. r« W.lcm." 


Fig. 4.6 

changes so that according to the curve 2, at = 15 , the ratio 
is .97, at 45 it is .91, at 105 it is .76 and at 180 it is .858. Curve 
1 is for a very great distance from the sphere; curve 4, for a dis- 
tance of 1 9. 1 cm. These curves are for one frequency * only and 
show the variation of intensity experienced by the observer at 
the point P' as he travels in a circle about 0. One of the inter- 
esting points is that the nearer the auditor to the speaker, the 
more advantageous is a position directly in front of the latter. 
* In Fig. 4.6 occurs the Greek letter "X," which represents "wave-length." 


Another interesting point is that there is a maximum * of inten- 
sity immediately in front of the speaker and one also immediately 

That the change in diffraction with frequency is marked is 
shown by Fig. 4.7 wherein the range of three octaves is consid- 
ered. The curves indicate that the higher the frequency, the 
more the advantage of the front position. It will subsequently 
be shown in Chapter IX that, in general, clearness of enunciation 
depends more upon the high than upon the low frequencies in 
the voice. The lower frequencies are more important in securing 
volume and the higher frequencies in securing clearness of speech. 
This fact should be given consideration in any use made of the 
results of the theoretical investigation. 

It is true that the head is not a sphere 60 cm. in circumference 
as the above theory assumes, but it is safe to conclude that the 
results of the investigation are applicable as an approximation 
in any consideration of diffraction about the head. 

4.9. Diffraction about the Head of an Auditor. — On account 
of an important theorem in acoustics called the "reciprocal" 
theorem, the conclusions of the foregoing theory can be trans- 
ferred to the case of diffraction about the head of an auditor. 
In such a case the source is assumed at the point P' and the 
relative intensities at A are calculated for various values of 8. 
Figures 4.6 and 4.7 are correct for this case also. It can then be 
said that the sound shadow or variation in intensity, produced 
by the sphere at a distance OP' with a source on the sphere, is 
the same as the shadow at points on the sphere produced by the 
source at P'. The term "shadow" is permissible, for since in 
optics a shadow is a definite variation in light caused by an ob- 
ject, so in acoustics we may speak of the variation of sound inten- 
sity produced by an obstacle as a "sound shadow." From the 
above mentioned curves we may reach the following conclusions: 

1. When listening to a distant sound of low frequency, say 

* A "maximum" intensity occurs at a point if at all neighboring points the 
intensity is less. But here the graph refers only to a variation in the angle 0. 



less than 200, there is but little to be gained by turning the head. 
(Here assume, for simplicity, that there is but one ear, but further 
examination shows the conclusion correct for binaural hearing.) 

2. The closer the source the relatively more important be- 
comes the position of the head. 

3. The higher the frequency the relatively more important is 
the position of the head. 






X^"** 1 *, 


*'240 n — " 


120 cm- 


r . 477 cm. 







.fcO^^^ 1 


V \ 




! ~T— - 




o .45 



15 30 45 60 75 90 105 120 135 150 165 180 
q __^ Degrees 

Fig. 4.7 

4. The shadow decreases with distance of source from the 

5. With 6 at 1 8o°, A and P' are on opposite sides of the sphere. 
The closer the source at P', or the less OP\ the greater is the 
shadow at A. This is indicative of the fact that an obstacle will 
cast a greater shadow the closer it is placed to the source. 

That obstacles do cast acoustic shadows can be demonstrated 
by using the tick of a watch and passing it about the head while 
one ear is closed, also in the lecture room by a highly pitched 
whistle, a sensitive flame (Section 15.14) and a small obstacle. 


4.10. Change of Quality by Diffraction. — From the above 
discussion it is evident that the music of a band in a city street 
will not have the same quality irrespective of the position of the 
observer. The higher frequencies are reflected more easily and 
are diffracted less easily than the lower frequencies. Conse- 
quently, because of the adjacent buildings, positions may be 
found where the high frequencies are relatively exaggerated or 
diminished. For the same reason, the quality of any music can 
be changed through the reflection from and diffraction about 
obstacles. The term "quality" is here used in a physical sense. 

4.1 1. Principle of Least Time. — Sound does not always take 
the shortest geometric path. It is one of the possible deductions 
from Huyghens' principle, Section 3.2, that the time required for 
sound to pass from one point to another is a minimum. That is, 
sound will reach the ear from a source in the least time. Thus 
the shell wave of a swift projectile passing overhead will reach 
an observer on the ground from a certain point in the trajectory 
determined by this condition of least time. In fact, it can be 
shown that if u is the velocity of the projectile, if v is the velocity 
of sound, the line drawn to the trajectory from the observer makes 
with this path an angle 6 such that the "cosine" of is equal to 
v/u. By "cosine" is meant the ratio of the adjacent side to the 
hypothenuse of a right angle triangle containing 0. 

The more general statement of the principle described in this 
section is that the time is either a maximum or a minimum. 
But the latter is the usual case and consequently the phenomenon 
usually bears the title given to the section. 

4.12. Passage of Aerial Waves about the Earth. — Krakatoa, 
a volcanic island between Java and Sumatra, was in violent erup- 
tion in 1883. On August 27th there was a culminating paroxysm 
and from this issued a wave of condensation which travelled out 
in all directions, passed about the earth and apparently again met 
at the antipodes to Krakatoa. It was then reflected, travelling 
back to the volcano, whence it returned in its original direction. 


The wave was detected by the change in the barometric pres- 
sure. It was observed issuing four times from the region of the 
volcano and three times returning. The actual velocity was of the 
same order of magnitude as the velocity of a sound wave in air. 


i. Draw two diagrams similar to Fig. 4.2, but assuming for one 
that «i is zero and for the other that #2 is zero. 

2. If the velocity of the wind, Fig. 4.2, varied not suddenly at 
00' but constantly with elevation, would there be refraction and why ? 

3. Assume that the velocity of the wind relative to the earth 
increases with elevation; show by drawing what influence the wind 
will have upon the direction of propagation and the wave front, if a 
plane wave of sound with horizontal wave front is emitted at the sur- 
face of the earth. 

4. By extending the discussion in the text prove more in detail 
the correctness of the statement that "the sound will be retained in 
this stratum very much as if transmitted between two parallel walls." 

5. Show that, when the temperature of the air is highest near the 
earth, the influence of temperature is to decrease the distance at 
which sounds may be heard. 

6. Show by drawing the possibility of refraction of sound from a 
source on the earth back to the earth again. 

7. Explain why "irregularities in the planity of the strata would 
be more effective in scattering the frequencies having the shortest 

8. How does the quality of sound from an airplane change with 
its passage? (A report on actual observation desired.) 

9. Does the refraction caused by wind or temperature vary with 
the frequency employed and why? 

10. If the medium is moving with a uniform velocity relative to the 
earth, what effect would this have upon the shape of a spherical wave 
from a point source, as observed from a stationary point on the earth? 

11. What is the justification for a reflector behind a speaker in 
the open air, as compared with indoors, and what effect would it have 
on the high and low frequencies? 

12. Cite cases in your own experience where quality is modified 
by diffraction. 

13. As you stand on the street in a busy city, in general how does 
the character of the noises from nearby sources compare with that 
from distant sources and why? 

14. Draw a diagram illustrating the effect of the elevation of 
church bells and justify the statement concerning the transmission 

15. Is it possible to determine from Fig. 4.2 the magnitude of the 
exaggeration (in the drawing) of the wind effect? 



S.i. Phase Change. — The terms " same phase " and " op- 
posite phase" were used in Sections 3.1, 3.2 and 3.7. The former 
refers to two vibrations of the same frequency in which the posi- 
tive maximum displacements occur simultaneously. But if one 
vibration is one-half of a period behind the other, they are said 
to be opposite in phase. But "phase," as in Section 3.7, is also 
used to indicate any difference in simultaneity whatever. The 
two vibrations are said to have a difference of phase or to be in 
different phases. Usually in acoustics only abrupt changes of 
phase occur and these are plus and minus half a complete vibra- 
tion or a difference of phase of 180 .* In this chapter will be 
discussed "reflection with change of phase" and "reflection with- 
out change of phase." The former refers to reflection wherein 
there is a change of phase corresponding to half a period; and the 
latter to a reflection without change of phase. 

5.2. Reflection without Change of Phase. — In Fig. 2.1, the 
waves from and 0' arriving at the point directly between these 
sources are in the same phased yet the resulting displacement is 
zero. By an extension of the reasoning involved, if instead of a 
wave from we have a wave front whose plane is parallel to the 
wall, the resulting displacement everywhere at the wall surface 
would be zero. This reflection can be simulated by removing 
the wall and substituting a plane I wave coming from the left 
that has (at the old position of the Vail) the same phase as the 

* The expression of phase in degrees is explained in Section 3.7. 

fNote that, although the displacements are actually opposite one another, 
yet in each case the relation of the direction of the displacement to the direction 
of travel of its respective wave is the same. This relationship always determines 
the phase. 

X A "plane wave" has a wave front that is a plane. 


original wave. The substituted wave is equivalent to the wave 
reflected from the wall. Since the wave from the left has the 
same phase at the wall position as the incident wave from the 
right, the reflected wave must be considered as having that same 
phase also. Such a reflection is called a reflection without change 
of phase. This phenomenon can be illustrated by the helix of 
Fig. 1.7. If a block of wood is placed at the end of the helix, 
the arriving wave does not give a displacement at the end but 
the combination of reflected and incident waves gives zero dis- 
placement. But the condensation of the incident wave at the 
block occurs simultaneously with the condensation of the reflected 
wave, making a pressure variation at the reflecting surface. 
Thus, when the reflected wave is in the same phase as the inci- 
dent wave, displacements are actually in opposite directions, but 
the two pressures are simultaneously positive or negative, mean- 
ing greater or less than normal pressure. 

This reflection without change of phase apparently occurs at 
every boundary where the second medium may be said to have 
relatively infinite inertia, i.e., may be considered to be rigid or 
immovable. Thus such a boundary may be a wall of solid mate- 
rial or it may be a liquid such as a body of water. It is true that 
the inertia is not infinite and that there actually is a sound wave 
of small intensity passing into the second medium, but we are 
neglecting this wave in the claim that the displacement at the 
boundary is zero. 

5.3. Reflection with Change of Phase. — The preceding sec- 
tion dealt with a reflection without change of phase, one in which 
the reflector was assumed to have infinite inertia. A case which 
is just the reverse will now be considered. Figure 5.1 represents 
an interface between a solid S and a gas G. If the sound wave 
comes from the right to the left, the solid has enormous inertia 
compared to the gas and the pressures in the gas are able to cause 
only very small vibrations in the solid. Neglecting these for the 
moment, it can be said that at the boundary we have total reflec- 
tion, and this without change in phase as explained in the pre- 


vious paragraph. Assume, however, that the wave is passing 
from left to right. It is then seen that the vibration in S is not 
impeded by G, because the gas has such a relatively small inertia 
and the vibration of S can occur approximately as if G were a 
vacuum. If G were really a vacuum, of course no sound energy 
could pass into it and there would be total reflection at the sur- 
face of S. With G a gas, some energy passes into it but only a 
small amount, for, as just stated, the vibration of S is approxi- 
mately independent of the presence or absence of G. This can 
be appreciated by comparing the mass per 
unit volume in S with that in G. At the 
surface the solid and the gas execute the 
same vibrations, but the energy in a sound 
wave depends not only upon the amplitude 
of the vibrations but also upon the mass 
in vibration; the greater the mass, the 
greater is the energy in its vibration. 
Since the density of the solid, S, is at least 
several thousand times the density of the 
gas, and it cannot transmit to the gas a 
greater amplitude than the solid possesses, 
the energy of vibration in the latter must 
be very much greater than that in the gas. 
then, to assert that but little energy is transmitted to the gas, 
and that the reflection may be regarded as approximately total. 
The relatively small inertia of G produces another condition. 
It can be shown mathematically that the pressure amplitude in 
G for a wave of given intensity would be very much smaller than 
for a wave of the same intensity in S. One might surmise, then, 
that with this practically total reflection the pressure amplitude 
of the refracted wave in G would be relatively small, indeed, very 
much smaller than that occurring in S with the incident wave 
travelling to the right or the reflected wave travelling to the left. 
But the excess pressure in G is at all times equal to the excess 
pressure in S at the surface, or to the sum of the excess pressures 
of the two waves in S at that point. Thus, these two pressure 

nXn \oA 

Fig. 5.1 
It is reasonable, 


waves in S are practically equal and opposite. Hence, there is 
a reflection with a change of phase of pressure also. 

It is interesting to compare the displacements. Since the 
waves in S are travelling in opposite directions, an opposition in 
phase causes the displacement amplitude at the surface to be 
twice the displacement amplitude of either wave. 

In the preceding section it seemed easier to determine the 
nature of the reflection by the consideration of displacements. 
In this section the pressure also is discussed. When a reflection 
is with change or without change of phase of the displacements, 
the same description must apply to pressures. 

Reflection with change of phase may be illustrated by the 
helix with a free end. Here the vibration completes its swing 
and is reflected back a half period after its arrival. 

5.4. Interesting Cases of Reflection in Gases. — It is not in 

general possible to determine the actual intensity of the reflectecl 
wave by a descriptive discussion. Hence a mathematical study 
must be made. The results of such a study in several cases will 
now be given. 

1. If the two media concerned are gaseous, the ratio of the 
displacement amplitude of the reflected wave to that of the in- 
cident wave, which meets the boundary perpendicularly, is 

„ . Vpi~ Vp p- ft /, t n 

Ratio = -7=—; — p or 717' W- 1 ) 

Vpi + V P * + ft 

where p is the density in grams per cu. cm. in the first medium and 
Pi in the second medium. The corresponding velocities are v and 
0i. In the case of a wave from hydrogen to air, p = .00008837 
and pi = .001276. Computation by (5.1) shows that the ampli- 
tude of the reflected wave is about 0.58 of the amplitude of the 
incident wave. In other words, the reflected energy is about 
(0.58) 2 or (0.34) of the incident energy.* 

a. If now we suppose the wave to be travelling in the opposite 
direction, that is, from air to hydrogen, the above fraction is 
negative, but of the same magnitude. This, the theory shows, 

* It can be shown that the energy of a plane wave is proportional to the 
square of the displacement amplitude, frequency constant. 


is to be interpreted to mean that now the displacements are oppo- 
site in sign, or that this is reflection "with change of phase" It 
is to be understood, then, that equation (5.1) will not only give 
the correct magnitude but will also state whether the reflection 
is with or without change of phase. From this equation we can 
conclude that if reflection occurs with the wave travelling from 
a more dense to a less dense gas, the reflection is with change in phase, 
but that if the propagation of the sound is from the rarer to the 
denser gas, the reflection is without change of phase. 

While the preceding discussion is accurate, it is not descrip- 
tive. If one will consider Sections 5.2 and $.3, he will observe 
that the conclusions therein would lead to the anticipation of the 
results just stated with gases. While one cannot extend the 
reasoning of those sections to gases of different density, yet they 
give a physical reason for the conclusions just stated concerning 
gases. There is, however, one fact that still lies hidden. A re- 
flection is either with or without change of phase. The change 
must always be either o° or 180 . There are no intermediate 
changes. It is to be observed, also, that in the broader state- 
ments of this Section the earlier limitation of equality of inten- 
sities of incident and reflected waves no longer holds. 

3. If the difference in the two media is one of temperature, 
there is reflection, but computation shows it to be slight. For 
example, suppose the change in temperature to be from 18 C. 
to 98 C. Then the velocities can be calculated by using equa- 
tion (1.3). If these two values of velocity be substituted in the 
latter part of equation (5.1), the ratio of the amplitude of the 
reflected wave to the incident wave is found to be .07 or 7 per 
cent. The reflection occurs with change of phase and the re- 
flected energy is only (.07) 2 or .005 or one-half of one per cent 
of the incident energy. Assume this conclusion applied to the 
case of a large stream of hot air from a furnace entering a room. 
It is seen that the reflection at perpendicular incidence from such 
a column of hot air is very small indeed. 

4. The effect of humidity may be considered by assuming 
sound passing from dry to saturated air. The latter is lighter 


by about one part in two hundred twenty. It can be shown by 
(5.1) that the reflected intensity is only about one eight hundred 
thousandth part of the incident intensity. 

5.5. The Image in Reflection without Change of Phase. — 

Referring to Fig. 2.1, the reader is reminded that the image 0' 
is in phase with 0. But reflection occurs without change of phase. 
This effect can produce a curious result. Imagine the source O 
in Fig. 2.1, with the image 0' in phase, to be brought gradually 
closer to the wall, where reflection without change in phase occurs. 
Then 0' must also approach the wall. When the distance be- 
tween and Q' is very small compared to a wave length of the 
frequency considered, they will act like two equal and like sources 
indefinitely near each other. The displacement one produces at 
any distant point (assuming the reality of f and the absence of 
the wall) is equal to and in the same phase as the displacement 
produced by the other. Hence the amplitude everywhere will 
be twice that produced by alone. This means that if a source 
is constant, it will, when placed very near a large reflecting 
wall, double the amplitude and quadruple the intensity every- 
where on its side of the wall. The total amount of energy in the 
hemisphere may be compared with that in the sphere (for the 
wave is spherical) when the wall is absent. The total energy in 
the hemispherical wave is twice the total energy in the spherical 
wave. Therefore the source of sound placed near the wall is 
caused to emit sound energy at twice its previous rate. While this 
discussion is correct, there is an assumption involved which is not 
made clear; namely, that the source at remains "constant." * 
The requirement of the constancy of the source limits the appli- 
cation of the conclusion concerning the doubling of the emitted 
energy, yet two points are made clear by the discussion. First, 
if the source is constant, the energy emitted is not always the 
same. Second, the surroundings may influence the amount of 
energy emitted from a source of sound, though the motion of the 

* A "constant" point source is one which injects and removes a fixed volume 
of air. A telephone diaphragm is a constant source if its motion remains constant. 


source, such as a vibrating reed or a telephone diaphragm, may 
remain the same. 

5.6. The Image in Reflection with Change of Phase. — From 
the preceding section and by similar reasoning, one can readily 
see that if the two sources are in opposite phase each will annul 
the other's effect. But this is the case in reflection with change 
in phase. Thus when reflection occurs with change in phase and 
the source is near the surface the intensity of sound in the inci- 
dent medium is greatly diminished by the reflection. This effect 
came strikingly to the attention of the physicists at work on 
submarine detection during the war. Since reflection in trans- 
mission from water to air is with change in phase, it would cause 
a diminution of intensity in the water in the case of any sub- 
aqueous source. The detector is of course placed in the water 
because even with this interference the intensity is much greater 
there. The velocity of sound in water is about 4.3 times that 
in air and hence the wave-lengths are correspondingly greater. 
The reader can readily show that the effect of interference just 
described is more pronounced at a given short distance of the 
source from the surface than would be the case if the velocity 
were that in air. 

5.7. Reflection at a Change in Area of a Conduit. — We have 
discussed the fact that a change in the character of the medium 
produces a reflected wave. But a change in the restrictions of 
the medium may also produce a reflected wave. For example, 
in Fig. 5.2, assume a conduit in which sound is being transmitted 
from the left to the right, and that at P there is an abrupt change 
in area of the conduit. There are two conditions at the point P 
which are fulfilled. First, the excess pressure at the point P is 
simultaneously the same for both parts of the junction, and 
second, the total flow of gas (in the vibration) is the same in the 
two branches. Strange as it may seem, the fulfillment of these 
two conditions requires the reflection of a sound wave at the 
junction. For the moment this reflected wave will be assumed. 


In the following section an explanation of its appearance will be 

It can be shown that if £ is the area of the larger tube in 
Fig. 5.2, and A that of the smaller one, the intensity of the re- 

• P 

Fig. 5.2 

fleeted wave divided by that of the incident one, or the percentage 
reflection, is correctly computed by the fraction, 

Thus, if the S is three times A, the amount reflected is 25 per cent. 
Let the same total alteration now be made in a series of many 
small steps instead of in one step as in Fig. 5.2. Then there will 
be a reflected wave at each small change in area. These reflected 
waves would not agree in phase. If there are many of them ex- 
tending over a length long compared to a wave-length, then there 
would be destructive interference among them and the resultant 
reflected wave would be small. Thus it happens that if the 
change in area is accomplished very gradually, extending over 
many wave-lengths, the transmission is practically undiminished. 
As a practical procedure, if reflection is to be avoided, it is essen- 
tial either to maintain the area of a conduit constant or to make 
the alteration gradually over a length of conduit containing a 
number of wave-lengths. In accomplishing the gradual altera- 


tion in area, it is not necessary that the inside surface of the 
conduit be without small abrupt changes in area. Thus a large 
number of small abrupt changes well scattered throughout a dis- 
tance long compared with the wave length will eliminate much 
of the reflected flow of energy. 

5.8. Cause of Reflection at a Junction. — One is entirely 
familiar with the reflection or rebounding of a tennis ball striking 
a wall. Yet it returns somewhat differently than an indoor base- 
ball or a ball of putty. This is a problem in mechanics and must 
include such considerations as momentum and elasticity. It is 
mentioned here to show that, after all, unless one is a student of 
mechanics, he is not familiar with the details of even this simple 
phenomenon with the tennis ball. Hence, in considering acoustic 
reflection, the rebounding ball is of no assistance. Indeed, it is 
actually misleading for reflection in the case of acoustics is a very 
different matter. Here we are not dealing with an object, but 
with a wave of changing physical condition. It must be recog- 
nized that the cause of a reflection of a wave must rest in the 
acoustic conditions at the reflecting surface. For example, con- 
sider the reflection of a plane wave incident normally on a rigid 
wall. In Section 5.2 it was shown that a plane wave will be re- 
flected, and specifically because of the condition required of the 
displacement at the wall. The resultant displacement at the 
wall, which is rigid, is zero at all times. But this is not the case 
in the incident wave, and consequently an incident wave only 
will not satisfy this condition of zero displacement. That condi- 
tion can be satisfied by an additional reflected wave with the 
actual displacements of these two waves equal and opposite at 
the wall. The existence of the reflected wave is thus shown to 
be a necessity. It does not come from a vibrating surface, but 
simply arises from the conditions at the wall. Of course in con- 
sidering reflection it is simpler to have in mind the approaching 
wave of pressure, which will produce a pressure at the wall, this 
pressure causing a return wave. But the important considera- 


tion at the moment is that the reflection occurs because the con- 
ditions there require an equal reflected wave. 

With this explanation perhaps the reason for the reflection 
of a wave at the point P in the conduit shown in Fig. 5.2 may be 
understood. Let us first assume that there is one wave entering 
from the left and one passing out toward the right. Assume the 
conduit infinite in length so there is no return wave. Let it also 
be assumed for the purposes of trial that there is only the one 
wave present. It was stated in the previous section that one of 
the conditions fulfilled at P is that there is but one value of pres- 
sure. If, then, there is but one wave entering and passing 
through the junction, it must retain its pressure amplitude un- 
changed. But this cannot occur. For the wave at the right of 
P is similar to the incident wave at the left of P except that it has 
a larger wave front. The wave at the right of P must therefore 
contain not only as much energy * as the wave approaching from 
the left but also an additional amount corresponding to the in- 
crease in area. But the mere presence of a change of area could 
not increase the flow of energy. That would be a violation of a 
general principle called the "conservation of energy." Clearly 
the assumption of but one wave is incorrect and some way must 
be found to meet the condition of one pressure value at the point 
P. As another trial assume that there is a reflected wave trav- 
elling to the left from the junction. It is well to notice that one 
cannot assume a second additional wave travelling to the right 
of P for there would then be two waves travelling in the same 
direction and these will be united becoming one wave. And it is 
not possible to assume a third additional wave on the right of 
the junction, travelling to the left, for this was eliminated by 
assuming an infinite tube. But two waves in opposite directions 
on each side of its boundary complete the possible assumptions 
even without the assumption of an infinite tube on the right. 
Hence, in the present case, the only additional wave possible to 

* It can be shown that the flow of energy per sq. cm. per sec. in a plane wave, 
and also a spherical one, varies with the square of the amplitude of the excess 


assume is a wave beginning at the area containing the point P 
and passing to the left. 

The problem is to determine if this assumed wave can meet 
the condition of pressure at the junction. Assume this reflected 
wave to occur and with change in phase. Then, as shown in 
Section 5.3, the pressures in the incident and reflected waves are 
opposite in sign. Since they are not of equal amplitudes there 
is a residue of pressure amplitude, and this pressure may be 
thought of as the source of the wave to the right. The assumed 
reflected wave can meet the condition of pressure at the junction. 
The excitation of the reflected wave and the fulfillment of the 
condition of pressure are two aspects of the same phenomenon. 
The fact that the wave has its origin without the vibration of a 
solid body need not disturb the student if he realizes that that 
is also the case when one whistles, when one plays the flute and 
when the wind howls about the corner. Space has been devoted 
to this discussion in order that the student may appreciate the 
essential difference between reflections in acoustics and reflections 
of a ball striking a wall. 

5.9. Reflection at an Open End of a Pipe. — Consider the 
end of a pipe as in Fig. $.3. At the point P we have the condition 
that the conduit opens out into the unconfined air where the 
pressure is normal. Obviously a wave passing to the right can- 

Fig. s-3 

not cause the entire region outside at the end of the pipe to 
experience as great changes in pressure as occurred in the wave 
within the tube. The reflected wave at P must then reduce the 
pressure at the opening, or it must furnish condensation for the 
rarefaction of the incident wave. This is reflection with change 
in phase, as we have previously seen. Another way of visualizing 


this result is as follows: at any point within the tube, displace a 
short length of the air column. The pressure ahead is thereby- 
increased and resists the motion. If this experiment is tried at 
the open end, the displaced air spreads readily out into the atmos- 
phere. The open end acts like the free end of a helix. As in that 
case, so here, the reflected wave increases the displacement and 
reduces the pressure amplitude. It is reflection with change in 
phase. One inherent difficulty in understanding why a reflection 
takes place at an open end of a pipe is that such a phenomenon 
seems at first thought not to be in accord with experience. A 
reflection from a wall is to be expected, but that from an open 
end is not so easy to anticipate. But, as will be shown in later 
pages, the action in such wind instruments as the cornet, trumpet, 
trombone and flute all depend upon the reflection of sound from 
an open end. This reflection builds up the intensity on the in- 
terior of these instruments and helps to make possible what is 
later described as "resonance." Such a reflection is common in 

There is considerable interest in the amount of reflection at 
the open end of a tube and this has been theoretically determined, 
but experimental data are lacking. If R is the radius of the tube 
and if the ratio of R to the wave-length X is small, then the per- 
centage of the wave travelling toward the open end dissipated 
into the open air is computed by the fraction. 

%TT 2 R 2 

X 2 

R i 
Thus if — is — , the amount dissipated is approximately 3 per 

R I 

cent. If — is — , the dissipation is approximately 19 per cent. 

A statement should be added to the effect that the theoretical 
investigation of the percentage dissipated assumes the open 
end of the tube to have a plane infinite flange. Although this 
is not in accord with Fig. 5.3 yet it is an approximation to the 
simple tube. 



In Section 4.9 a certain reciprocal relation was mentioned. 
This relation, when applied to the case above, states that when a 
source of sound is placed in a tube with an open end P, the inten- 
sity at a given outside point has the same value as would be 
observed were the positions of source and point of observation 
interchanged. That is, if but little sound leaves the tube, but 
little would enter were the source on the exterior. In fact, a tube 
with a small opening does not supply a good means of egress of 
the sounds. It is likewise a poor receiver. These considerations 
explain why the receptive qualities of a tube need to be improved 
by flaring the open end. 

5.10. Reflection at a Closed End of a Pipe. — At the closed 
end of a pipe the reflection occurs just as from a wall. It is re- 
flection without change of phase. 

S.i 1. Total Reflection at an Interface. — Assume that we have 
a sound wave of velocity v passing into a second medium and 
therein having a greater velocity v'. As in Fig. 4.1, the direction 
of the wave is changed and 
the angle measured between 
the perpendicular to the sur- 
face and the direction of the 
wave changes from to a 
larger angle 0', as in Fig. 5.4.* 
As explained in Section 4,2, 
if now one increases the 
"angle of incidence," 0, the 
"angle of refraction," 0', in- 
creases even more rapidly 
until finally the refracted 
wave just skims the surface, 
i.e., 0' is 90. If now be 
increased still further, there is no refracted wave, total reflection 
ensues, and we have only the reflected wave, which is not drawn. 
The minimum angle of incidence at which total reflection occurs 

* The reflected wave is not shown in Fig. 54. 

Fig. 5.4 


is called the "critical angle." It is clear that the greater the 
difference in velocities of the sound wave the smaller the angle 0, 
at which total reflection begins. Thus, in passing from air into 
water, this angle has been computed to be approximately 13. 5 . 
In the case of a column of hot air such as that described in Sec- 
tion 5.4, with the velocities having a ratio of 1. 15, 6 is approxi- 
mately 6o°. This phenomenon of total reflection can be repro- 
duced in the laboratory and may often occur in experience. But 
if the refracted wave has a very small intensity as in the trans- 
mission from air to water, its presence or absence is not noted 
and the phenomenon of total reflection is not appreciated as such. 
Again, if the angle 6 is large, the observer may think the sound is 
coming directly from the source and fail to recognize the fact of 
total reflection. Many reports have been made of apparent total 
reflection caused by unusual meteorological conditions. It should 
be observed that the refracted wave may cease to exist, but that 
the reflected wave is always present. 

5.12. Absorption along a Conduit. — If a conduit has a rigid 
non-absorbent wall obviously the wave passes along it without 
being scattered. The wave front will remain plane as the wave 
travels along the tube. But it must not be thought that the wall 
of the tube plays no part in this transmission. Suppose it were 
made of felt. Then the wave of pressure travelling along the tube 
would cause motion of the air in the felt, producing absorption. 
Indeed, this is a method used in preventing the sound passing 
through a large ventilating flue. It is observed that although 
the wave may be said to travel along the tube there is nevertheless 
a divergence into the walls and a marked absorption. 


1. If there are air currents in an auditorium, or if there are some- 
what sharp variations in temperature, and if these conditions are not 
constant, what effect would be had on the sound intensity at any 
given point, assuming a constant sound source? 

1. In a previous chapter was presented the phenomenon of ab- 
sorption in the walls of a room. In the light of the present chapter, 


assuming the velocity of sound in the walls to be much greater than 
in the air, what additional phenomenon would you note as occurring 
at the walls in a room? 

3. Justify the statement, "That the effect of interference just 
described is more pronounced at a given short distance of the source 
from the surface than would be the case if the velocity were that in 

4. In discussing the effect of change of area in a conduit the state- 
ment was made that " there would be destructive interference among 
them," etc. Show why this interference occurs. 

5. What is the one condition depending on the physical property 
of the incident medium that determines the possibility of total reflec- 
tion at an interface? 

6. Which discussion is more general, that relating to Fig. (4.1) or 
that relating to Fig. (5.4) ? Explain. 

7. When like waves are travelling in opposite directions and their 
pressures at a point are always equal, what is the relation between 
the displacements? Justify the answer. 

8. At a rigid boundary, the reflection occurs with or without 
change of phase of displacements? 

9. When the phrase "with change of phase" has been used in the 
text, to what does it usually refer, pressure, velocity or displacement? 

10. Can one hear a distant sound better if he listens at the surface 
of an "infinite" wall, and why? (Assume that the ear detects pres- 
sure and not displacement.) 



6.1. General Phenomenon of Resonance. — The term reso- 
nance, when used in a broad sense, refers to the excitation of a 
vibration in a body by a wave from another sound source. The 
phenomenon appears most striking when the frequency of the 
initial wave equals the frequency of the natural vibration of the 
body caused to vibrate. For example, the air in an empty globu- 
lar vessel may be made to speak loudly if its critical tone is 
sounded in the same room. There is an impression that the phe- 
nomenon of resonance provides a method of multiplying or am- 
plifying the flow of energy after it has become sound energy. But 
this is incorrect. Resonance accomplishes, in general, two not 
altogether different results. When the resonator or the resonat- 
ing body is at a distance from the original source such that the 
vibration of the latter can to no appreciable extent be affected 
by the sound wave from the former, then the total emission of 
sound from the original source is constant. In this instance the 
resonator does not increase the flow of sound energy or the flow 
of energy at the resonator. But there are cases to be discussed 
later where the resonator affects the source, for example, sounds 
from the vocal chords. In such a case there is a difference in the 
intensity of sound given off by the source when the resonating 
body is present. Hence one can say that resonance may increase 
the flow of energy which is becoming available as sound but it 
never can multiply the flow of sound energy already present. It 
may succeed in storing up energy over a period of time and thus 
make the final effect more powerful. This is true of a Helm- 
holtz resonator described in Section 6.6. Or resonance may ac- 
tually succeed in causing a source of sound such as a vibrating 
reed, a vibrating string, or the vibrating vocal chords to emit 
more sound energy than would be the case in the absence of reso- 



nance. In both cases the casual observer would claim that the 
amount of sound energy was increased. 

As will be seen, resonance is not only of great practical impor- 
tance but also of intrinsic interest. In order that the term "reso- 
nance" may be used with sufficient freedom, its meaning should 
be considered to embrace mechanical vibrations of all kinds. 

6.2. Plane Stationary Waves. — There is a phenomenon, i.e., 
"stationary waves," described in all elementary texts in acoustics. 
The reader is reminded that no "wave" in the sense first used in 
this text could remain stationary. That the use of the word 
"stationary" is nevertheless appropriate will now be shown. 

If two plane waves of the same amplitudes and frequencies 
are sent over the same path, they will combine, and if in the same 
phase they will give a wave of twice the amplitude of either. If 
they are in opposite phases they will completely annul one an- 
other. If, however, the waves are travelling in opposite direc- 
tions over the same path, the resultant effect cannot be either of 
those just specified. What occurs is not easy to visualize for the 
waves are moving in opposite directions. One can understand the 
result most quickly by an analogy. If one fastens horizontally 
a long helical spring of small diameter at 0, and grasps it at the 
end A> he can, by moving the hand up and down, send a series 
of transverse waves from A to 0. The waves sent out will be 
reflected from and, if the vibration at A is maintained, the 
resulting motion of the spring will be a combination of the two 
waves. By properly adjusting the frequency of motion at A, the 
resulting motion may be as in Fig. 6.1. The point A in the figure 
is represented at rest because when this exact frequency is se- 
cured, the motion of the hand is small in comparison with the 
motion of the spring. In (i) the spring vibrates between the 
position AaO and AbO. The reason an exact frequency must be 
used at A is that one must produce an interference such that A 
and will remain at rest, and yet such that the other points along 
the spring will have motion. If now a more rapid and yet appro- 
priate transverse vibration of A is tried, (2) or (3) will be secured. 



The appearance of the vibration is similar to that in (i), save that 
instead of one segment we have two and three respectively. Now 
the resultant variations as shown in Fig. 6.1 could have been ob- 
tained graphically instead of by experiment. The graphical 
method would have been to superimpose two such drawings as in 
Fig. 1.9, causing one to move to the right and the other to the 

Fig. 6.1 

left with the same speed, and then adding the displacements rep- 
resented by the two graphs at different instants. The graphical 
method must of necessity give the same resultant at succeeding 
instants as occurs with the spring in the actual experiment or as 
shown in Fig. 6.1, for the graphical method is essentially like the 
experiment itself. In the uppermost drawing in Fig. 6.1, the 
curve a and the curve b of course represent the extreme positions 
of the spring which will occur at times differing by half of that 
required for a complete vibration. The dotted curves represent 
the positions of the spring at selected other intermediate instants. 
It has just been stated that Fig. 6.1 could have been obtained 
by a graphical method as well as by experiment. Consider what 
steps would need to be taken to determine the resultant of two 
longitudinal waves of equal amplitude and frequency travelling 
in opposite directions. First, two curves similar to Fig. 1.9 would 


be prepared, the vertical distances now representing longitudinal 
displacements. Then these curves would be displaced in opposite 
directions and at each selected instant the two displacements 
everywhere added. The resulting curve for any instant would 
indicate the resultant longitudinal displacement at every point. 
But the preceding paragraph states that two such curves treated 
in that manner will give Fig. 6.1. Hence one will obtain similar 
final curves, but in one case having in mind transverse waves and 
in the other longitudinal waves. Fig. 6.1 may thus be taken to 
represent longitudinal waves as well as transverse waves. It will 
then be assumed that the above three figures, (1), (2) and (3), 
may in each case represent two longitudinal waves of like fre- 
quency travelling in opposite directions, the vertical distances 
being proportional to horizontal displacements. 

It is now possible to discuss what is called a "stationary" 
sound wave in air. It is evident that at points A, c y d and 0, 
there is approximately zero displacement. The medium is "sta- 
tionary," at those points. But consider the direction of the dis- 
placements in the neighborhood of c and d. They are first toward 
these points and then away from them. Hence the medium at 
c and d experiences changes in pressure. The same is true of A 
and 0. When A and d in (3) are points of condensation, c and 
are rarefactions, and vice versa. These points of no motion 
are called "nodes." At the midpoint between A and c the me- 
dium has the greatest displacement. This midpoint is called a 
"loop." It has just been stated that when A is a condensation, 
d is also. The distance from A to c , or c to d> ordtoO is one-half 
wave-length. Another interesting fact should not escape atten- 
tion. In an ordinary progressive sound wave at any instant the 
phase differs from point to point along the wave. In a stationary 
wave the phase in one segment such as Ac is everywhere the same 
the displacements of the particles in this segment differing only 
in amplitude. But the phase in one segment is at the same in- 
stant opposite to the phase in the adjacent segment. At a loop, 
the adjacent particles have virtually the same displacements and 
hence there is no change in condensation or pressure. 


6.3. Stationary Waves in a Cylindrical Pipe Closed at One 
End. — In the previous chapter it was shown that one can obtain 
reflection with or without change of phase. In the case of per- 
pendicular incidence upon a wall where the reflected wave passes 
in a direction opposite to the incident wave, and where the ampli- 
tudes of the two waves are equal the conditions for stationary 
waves exist. 

Consider a cylindrical pipe AB, Fig. 6.2, and assume that 
plane waves 00' from a distant source of sound enter this pipe 
and are reflected at B. If we assume that this reflected wave B 


Fig. 6.2 

to A does not suffer a reflection at A, but goes out into space, we 
will have in AB the condition of two like waves travelling in 
opposite direction or the condition for stationary waves. The 
reflection at B will be without change of phase, there being no 
displacement but variation of pressure only. B is therefore at a 
displacement node. Furthermore, in the distance from B to A 
there will be a node at each one-half wave-length, and between 
them, or at one-fourth, three- fourths, five-fourths, etc., wave- 
lengths, will appear a loop. 

So far we have assumed no reflection at A . We now realize 
that if it is possible to have a stationary wave in a pipe closed at 
one end and open at the other, there may be a certain position 
for A y the open end of the pipe, which will not only permit but 
encourage the stationary wave to exist. The reflection at A of a 
wave travelling to the right, here called Ri, is with a change of 
phase, as shown in Section 5.9. Ri and its reflection at A called 
i? 2 , would, if the latter were equal to the former, form a station- 


ary wave with a loop at A. In fact they both conspire to this 
end. This suggests that we consider the idealized pipe in Fig. 
6.2 cut off so that A will be at a displacement loop. Then from 
A to B is an odd number of quarter wave-lengths. Since the 
reflection at B is without change of phase, then from A to B and 
back to A would be equivalent to a path of twice this length and 
the wave Ri when incident at A would be an odd number of half 
wave-lengths (or twice an odd number of quarter wave-lengths) 
ahead of the incoming wave at A. That is, it is out of phase with 
the incident wave. Then R 2 , because the reflection is with change 
of phase, would be in phase with the incoming wave. Thus all 
three waves at the point A would have the necessary phase rela- 
tions to form a displacement loop. The only other condition to 
be met for a stationary wave is that the combined amplitude of 
R 2 and the incoming wave would equal the amplitude of Ri. This 
requires an entering wave that has a flow of energy precisely 
equal to the difference between the corresponding values of Ri 
and R 2 . But this means that the flow of energy from the tube, 
or the energy of Ri less that of i? 2 , is equal to the energy flowing 
into the tube. This is clearly the condition for steady operation, 
without any change in energy in the tube. With a steady source, 
this is of course a possible condition. Hence cutting off the pipe 
at one of the original loops places A at the correct point for a 
possible stationary wave. Assume that we have a source and 
cause a wave from it to enter a cylindrical tube of the length 
above described. The stationary state will not be attained at 
once, for a portion of the wave from B does not get out of the 
pipe but is trapped by reflection at A. Consequently the acoustic 
energy within the pipe will increase with time until the rate the 
energy is dissipated is just equal to the rate of influx. 

By this process the energy per unit volume within the pipe 
region has been made much greater than if the pipe were absent. 
But this energy has not been created by the pipe, but rather stored 
over a period of time. It is to be observed that the agreement of 
the reflected wave at A with the incoming wave indicates that 
we are using the frequency which may be called a "natural" fre- 


quency of the pipe. It is the frequency with which the interior 
would oscillate after the volume of air is given a sudden blow. 
The phenomenon whereby we build up the intensity by using the 
natural frequency is called "resonance," though the term is not 
limited to such a case, as was stated at the beginning of this 

One of the simplest ways of showing such a case of resonance 
is (see Fig. 6.4) by placing a vibrating tuning fork over a cylin- 
drical jar in which the volume of air has been adjusted by the 
water content, so that its frequency is the same as that of the 
fork. The building up effect can be illustrated by varying the 
length of time the fork remains at the opening of the resonator. 
For short intervals, the longer the time, the greater the intensity. 

6.4. Resonance. — In any case of resonance the greatest effect 
is obtained when the frequency of the stimulus is approximately 
equal to the natural frequency of the vibrating body. But we 
may also get an increased effect if the natural frequency is not 
so nearly matched. The apparatus shown in the accompanying 
drawing, Fig. 6.2, will illustrate this point. A disc, D, is sus- 
pended by a spring. If the disc is " offcenter," or 
if there is added to the disc along a radius a small 
weight PF, then in each revolution of the disc 
there will be an impulse given the spring both 
upward and downward. The experiment is per- 
formed in the following manner. With its axis 
held stationary, the disc is given a spin. The 
axis is then released. At each revolution the disc 
gives the spring the impulses already mentioned, 
but these do not succeed in giving the spring a 
marked oscillation until, as the angular velocity 
decreases, the natural period of oscillation of the 
combined spring and suspended weight is ap- 
proached. Then the vertical oscillation becomes 
evident, increasing until it is very vigorous at the agreement of 
the frequency of rotation, the natural one. Then the oscillation 



decreases to a small value. This shows that while the maximum 
effect is obtained at a certain rotational frequency, the phenome- 
non of resonance is apparent at adjacent frequencies also. 

6.5. Emission of Sound Increased by Resonance. — If an 

experiment is performed with a cylindrical tube and a vibrating 
tuning fork as in Fig. 6.4, it is found that when the air column 
has for its natural frequency that of the fork, there occurs the 
maximum emission of sound from the two. But since the cylin- 
der cannot give out more or less energy than it receives, assuming 
no dissipation within, the emission of energy from the cylinder 
must be identical with the energy 
flowing into it from the fork. Hence 
we must conclude that when placed 
above the resonating cylinder the 
fork emits energy at a greater rate 
than when alone. This illustrates 
an important fact not usually under- 
stood, namely, that resonance does 
not create energy but may make pos- 
sible a greater emission from the 
source. This increase in output can 
be explained by the phase relation- 
ship of the velocity and pressure 
at the source. An analogous case is setting a swing into vibration 
by pushing when at the midpoint of the arc or at the maximum 
velocity. The push and the velocity are in the same phase. It 
may be remarked that cases may arise in acoustics where the 
phase relationship is unfavorable rather than favorable. In such 
an event, the emission of sound is made less by the presence of 
the resonator near the source. 

That in the preceding experiment a fork has been caused to 
emit energy at a higher rate is based not only upon theory but 
also upon experiment. The literature records experiments by 
Koenig wherein a fork sounded about 90 seconds without a reso- 
nator and 10 seconds in the presence of one. Obviously the 

Fig. 6.4 



change that occurs depends upon the internal losses of energy in 
the fork and upon the dimensions of the resonator. Thus one 
should not expect the same relative change in all similar experi- 

The phenomenon of causing a vibrating source to give off 
energy more rapidly because of the presence'of resonance is shown 
in practically all musical instruments including the vocal chords. 

6.6. Resonance in a Volume Having an Orifice. — If, instead 
of a cylindrical pipe, we have a changing cross section, the theory 
becomes very difficult. In fact, the method then utilized is only 
an approximation. An illustration will be made of a volume 
containing an orifice as in Fig. 6.5. Such a volume and orifice is 

commonly called a "Helmholtz reson- 
ator." If this volume is set into 
vibration in its natural frequency, the 
maximum particle velocity (see Sec- 
tion 1 . 1 2) will occur at 0. V is so large 
that the motion inside is slight. The 
volume thus acts like a cushion or a 
spring, the displacement at the orifice 
causing a compression (or rarefaction) 
throughout the interior. In the chan- 
nel at the orifice, however, the motion is relatively violent and the 
mechanical forces arise not from condensation or rarefaction but 
from the rapid acceleration or rate of change of velocity of the 
mass of gas in the channel. In short, the predominant physical 
factor in the V region is elasticity and in the region, inertia. 
We can therefore make this resonator analogous to a spring and 
a weight as shown in Fig. 6.J ffl. In mechanics it is shown 
that the frequency of vibration of such a spring is 

1 1 fk 

where m is the mass of the weight, k is the so-called "constant" 
or stiffness of the spring, n is the frequency and T is the period. 

Fig. 6.5 


In acoustics an analogous equation is derived, and is found to be 
as follows: 

1 a fc 

wherein a is the velocity of sound, c is called the "conductivity" 
of the orifice and V is the volume of the chamber. All distances 
are measured in cms. If the orifice is a circular hole in a thin 
wall, the value of c is two times the radius. To illustrate the 
formula, compare the case of a cylinder closed at one end (as in 
Fig. 6.4) and having a natural fundamental frequency of 200 
cycles, with that of a Helmholtz resonator, Fig. 6.$. Let the 
cross sectional area of the cylinder be 10 cm. 2 Its length would 
of necessity be one-fourth that of the wave-length of 200 cycles 
or J (33200 -5- 200), or 41.5 cm. This is the necessary height * 
of the cylinder resonating with a frequency of 200. If a similar 
cylinder, having a top closed, excepting a circular orifice 0.5 cm. 
in diameter, is used as a resonator, it is readily shown by the. 
above formula that the height of this cylinder need be only 36 
cm. in order to have 200 as a natural frequency. 

If instead of a simple orifice for this Helmholtz resonator there 
is provided a neck 2 cm. long and 0.5 cm. in diameter, the natural 
frequency of 200 cycles will be obtained with a length of cylinder 
of only 4.9 cm. The formula for conductivity of such a neck is 
as follows: 


C = L + ^' 


where R is the radius and L the length of the neck. This is a 
more general value of the "c" which may be used in the formula 
for the frequency of a Helmholtz resonator. 

The above example illustrates the possibility of securing low 
natural frequencies but with relatively small volumes of air. An 
illustration occurs also in the resonance of the human voice. A 

* There is an "end correction" for every open pipe which is not considered in 
this example. It is discussed in Section 6.9. 


deep bass note sung by a man would require a 10 foot open organ 
pipe to produce. 

6.7. Resonance of the Voice. — In the case of the voice, the 
resonance cavities are found in the larynx, pharynx, mouth and 
nasal passages. To what extent the sphenoid sinuses and the 
right and left antrums may enter into the resonance of the voice 
is not known but is presumably small. The trachea below the 
vocal chords is not in a position to emit sound successfully and 
therefore its resonance is not of serious moment except as it may 
influence the emission of energy from the source. The phrase 
"throwing the voice" can scarcely refer to anything other than 
the control of quality by modifying the resonance through changes 
in the positions of the tongue and palate. This is the method 
used in producing different vowel sounds. The preceding for- 
mula assumes that the resonating cavity and the orifice are well 
defined. So long as this is true the shape of the volume is not of 
any significance* But in the mouth and pharynx cavities the 
volumes and orifices are less clearly differentiated, and conse- 
quently the shape of the cavities is a factor. The skill acquired 
by the individual in varying the resonance properties of the 
mouth, pharynx, nasal passages and larynx by variations in the 
two first named is indeed remarkable. 

6.8. Resonance in Cylindrical Pipes. — In Section 6,^ y it was 
shown that stationary waves may exist in a pipe and that there 
is a displacement loop at the open end and a displacement node 
at the closed end. Assume a pipe of length L open at both ends. 
The stationary waves formed in such a pipe must have a dis- 
placement loop at each end. The lowest frequency must have 
but one displacement node on the interior of the pipe. The suc- 
cessive higher frequencies must have two, three, four, etc., nodes. 
Thus it is simple to show that the lowest resonating or natural 
frequency is one having a wave-length 2L, for a wave-length is 
the distance between alternate nodes or alternate loops. The 

next highest frequency has a wave-length of L, and the next -£. 


The corresponding frequencies as multiples of the lowest or fun- 
damental are 1, 2, 3, 4, etc. Thus the natural frequencies of an 
open pipe contain all integral multiples of the fundamental fre- 

The case of a pipe closed at one end is somewhat different. 
Here there is a node at the closed end and a loop at the other. 
Consequently the lowest frequency has a wave-length of 4L, or 
four times the length of the pipe. The higher frequencies have 

wave-lengths of - L y - L, - L, etc. The natural frequencies of the 

o J I 
pipe are thus, in terms of the fundamental, 1, 3, 5, 7, etc. The 
natural frequencies contain only odd integral multiples of the 
fundamental frequency. The difference in natural frequencies 
between an open pipe and a pipe closed at one end is of impor- 
tance in wind instruments. 

In the construction of the nodes and loops as in Fig. 6.1 (3), 
one must bear in mind that such graphs do not pictorially repre- 
sent the longitudinal stationary waves in the pipe. The dis- 
placements are in the direction of the axis of the pipe, whereas 
the displacements in the graph are perpendicular thereto. 

6.9. End Correction of an Open Pipe. — The relationship be- 
tween the length of a pipe and the wave-length of a resonating 
frequency as stated in the foregoing section is not exactly correct. 
The pipe cannot be considered as ending, in an acoustical sense, 
at the opening. The reason is that the wave at this point cannot 
at once spread out into space. There is an additional equivalent 
length which is definitely related to the "conductivity" men- 
tioned in Section 6.6. One of the latest determinations of the 
correct value for the additional length is that of Bate.* He states 
that the correction for the open end of an open-organ flue-pipe 
is 0.66 times the radius of the opening and that the value is inde- 
pendent of the frequency, at least over the octave used in the 
experiments. To the actual length of the pipe represented by L 
in Section 6.6 must be added a length equal to 0.66 times the 
radius of the opening. 

* A. E. Bate, Philosophical Magazine 10, 65, 917, 1930. 

9 2 


6.10. Resonance in Conical Megaphones. — When the mega- 
phone is used as a receiver, its resonating frequencies are not like 
a cylindrical pipe, closed at one end, but, strangely enough, they 
are the frequencies that would be obtained in a cylindrical pipe 
having the same length but open at both ends. The fundamental 
of the conical horn is therefore twice the fundamental of a cylin- 
drical pipe of the same length closed at one end. The resonance 
of a conical horn differs very much from that of a cylindrical 
pipe. The latter will first be described. Fig. 6.6 shows the effect 

of resonance in a cylindrical pipe open at both ends, the source of 
varying frequency being held near one end. The vertical height 
in the graph indicates the intensity in the pipe, assuming the 
source of varying frequency always to produce the same quanti- 
tative changes in condensation and rarefaction at the opening. 
The frequencies are indicated as being I, 2, 3, or 4 times the fun- 
damental frequency, "/." 

It is to be observed that the intensity is reduced practically 
to zero, but more exactly to no amplification, in between each 
resonance frequency, and that the intensity at the successive nat- 
ural frequencies, 2, 3, 4, X /, decreases very rapidly. In point of 
fact, the intensities decrease much more rapidly than is here 
shown. The conical horn has a very different effect. Fig. 6.7 
is a similar experimental graph by Stewart * for a conical horn. 

* Stewart, Physical Review, XVI, Oct. 1920, p. 313. 


(Actually, the experiments were performed with changing lengths, 
frequency constant, but one can show by theory that the two 
curves would be alike.) These figures show a striking difference 
between resonance in a pipe open at both ends and a conical tube 
closed at the vertex. With the latter, the intensity at resonance 
does not decrease rapidly with increasing frequencies but the 
maximum values are approximately equal. Moreover, the in- 
tensity does not become small at intermediate frequencies. Sig- 
nificantly, the minima increase with increasing frequencies. This 

Fig. 6.7 

is the secret of the fact that a conical horn acting as a receiver 
will amplify the emission of sound energy any frequency that is 
high compared with its fundamental. Thus a long conical horn 
will amplify all the tones existing in almost any sound and will 
therefore give a fairly faithful reproduction of the quality or 
timbre of the original sound. This is an important consideration 
in the construction of trumpet receivers and reproducers. 

In considering the action of the conical horn as a transmitter, 
one should recall that a resonator can increase the emission of 
energy from a given source. Moreover, as indicated in Section 
3.9, the area of the large opening is a wave front and consequently 
the horn can direct the sound or partially concentrate it in a 
favored direction. The conical horn is successful as a transmitter 
because of these two facts and because of the nature of its ampli- 
fication as shown in Fig. 6.7. 


In connection with the above, it should be emphasized that 
the megaphone does not ordinarily serve as a concentrator of 
sound but as a resonator. Of course if the frequency becomes 
very high the sound can be reflected somewhat as is light, but 
sounds usually encountered should not be considered as capable 
of concentration by a conical horn. The fallacy in the concen- 
tration idea can be observed experimentally as follows. If one 
holds a megaphone to the ear and listens to a sound having a fre- 
quency half-way between " i " and "o," Fig. 6.7, he will find that 
the intensity is practically the same with or without the mega- 

6.1 1. Megaphones not Conical. — The phonograph or loud 
speaker horn is not conical but may be considered as made up of 
fulcra of horns of different angles and lengths. Hence it will 
have many resonating frequencies, but none are as marked as 
those occurring in the conical horn. A very common shape is 
that of the exponential horn. Here the diameter increases rap- 
idly forming a flare. The harmonics can be computed. Mention 
will be made in Chapter XV of the use of horns in loud speakers. 

6.12. Stationary Waves in General. — It must not be sup- 
posed that in all stationary waves the adjacent nodes are one- 
half wave-length apart. For example, in a conical horn closed 
at one end, the fundamental vibration has a node at the closed 
end and a loop at the open end, and, according to the discussion 
of stationary waves in this chapter, the distance between this 
node and loop should be one-quarter of a wave-length. Yet it 
has just been stated that the horn length is one-half of the wave- 
length of the fundamental frequency. The apparent contradic- 
tion is explained by an incorrect inference from the preceding 
discussion of stationary waves. The displacement nodes and 
loops were formed by two plane waves of equal frequency and 
amplitude travelling in opposite directions. Other more compli- 
cated cases of nodes and loops will be later discussed. No general 
statement can be made concerning the distance between a node 


and a loop. For such plane waves the distance between a dis- 
placement node and loop is one quarter of a wave-length. This 
is not the case in a conical horn for there the waves are spherical. 

6.13. Resonance in Musical Instruments. — In stringed in- 
struments the effort is made to get the energy from the strings 
to the atmosphere. On the violin this is done by the rocking of 
the bridge which sets the body of the violin in motion. Thus the 
surface exposed to the air is increased. Moreover, the air cham- 
ber within has a multitude of natural frequencies which in turn 
increase the emission of sound. This phenomenon of increase of 
emission by resonance has already been discussed. The function 
of the sounding board on the piano is similar to the body of the 
violin as has been described. The application of the two prin- 
ciples, (1) of increasing the area exposed and (2) of utilizing reso- 
nance, may be made in the explanation of the effect with any 
stringed instrument. A consideration of the possibility of reso- 
nance in the solid body of the violin or of the sounding board is 
omitted at this point but will be subsequently mentioned. 

In wind instruments, the tone is produced through resonance. 
The quality of sound produced when a wind instrument voices 
any given fundamental depends upon the possible resonating fre- 
quencies, the presence of these in the source of sound, and the 
rate of emitted energy of each of these frequencies by the source. 
A difference in resonating frequencies and in the sources of the 
sound will merely be illustrated here, a more complete presenta- 
tion being reserved for Chapter XV. The clarinet is a pipe 
practically closed at one end by a reed. Its resonating frequen- 
cies are therefore, 1, 3, 5, etc., times that of the fundamental. 
On the other hand, the oboe and the bassoon are conical tubes 
closed at the vertex by a reed. The resonating frequencies are 
1, 2, 3, 4, 5 times that of the fundamental. 

6.14. Resonance in Buildings. — There may be found reso- 
nance frequencies in any small room or recess. The organ builder 
is well aware of this difficulty and adjusts the intensity of his 


pipes to produce the correct proportion of intensity for the organ 
in its final position. 


1. Draw a curve representing a stationary wave at the time of 
maximum displacement. Indicate the direction of the displacement, 
the points of condensation and rarefaction. 

2. Describe the conditions existing at one-fourth of a complete 
vibration later than in the preceding question. 

3. What is "stationary" in a so-called stationary wave? 

4. What is peculiar about "phase" in a segment of a stationary 

5. What are three conditions necessary for a stationary wave such 
as discussed in this chapter? 

6. In the three examples in Fig. 6.1 what are the relative fre- 

7. Give a broad meaning of resonance. 

8. What are the two possible fundamental accomplishments of 
resonance? Give illustrations of each. 

9. From the discussion concerning conical horns, what would you 
anticipate concerning the effectiveness of a hearing trumpet 30 cm. 
long, and why? 

10. The theory of resonators herein given does not discuss the 
nature of the material of the walls. Under what condition is the 
omission justified? 

11. Why does it require time to build up the energy in a pipe 
closed at one end? 

12. Why is it possible to build up energy in a pipe open at both 

13. Show that in the resonator, Fig. 6.5 (*), if the radius of the 
orifice is increased, the frequency is increased, and that if the length 
of the neck is increased, the frequency is decreased. 

14. In the experiment illustrated in Fig. 6.4 what limits the inten- 
sity of sound produced? If the chamber were lined with absorbing 
material would the same intensity be produced? 

15. Would it be possible for a man to play such a note on a musical 
instrument that a large steel bridge would be set into vibration and 
finally collapse? 

16. Construct the graphs verifying the resonance wave-lengths of 
cylindrical pipes as given in Section 6.7. Compare the physical action 
with the graphs. 

17. If one fills his lungs with hydrogen gas and then attempts to 
speak in the usual way, a marked effect is produced on the voice. 



7.x. Musical Tones. — It is not the purpose to enter here 
into either a physiological or a psychological discussion of the 
requirements for a tone that is "musical." It is a fact, how- 
ever, that we demand for the most pleasing consonance a simple 
ratio between the frequencies of a complex musical sound. This 
is illustrated by the development of our musical instruments 
which in general give tones having the ratios of frequencies, 
1:2:3:4:5, etc. 

7.2. The Vibration of a String. — In a string we are usually 
concerned with transverse vibrations. It is true that longitu- 
dinal vibrations can be set up in a string or in any solid. A 
piano string, if rubbed so as to excite longitudinal vibrations, will 
give a very high tone. In all string instruments the transverse 
and not the longitudinal vibrations are utilized.* The longitu- 
dinal vibrations are displacements along the direction of wave- 
propagation and consequently possess condensations and rare- 
factions.f The velocity of such waves depends upon the force 
necessary to cause a given condensation and the density or mass 
per unit volume of the material. This dependence has been dis- 
cussed in Chapter I. In the case of a steel wire the velocity is 
about 5,000 meters per second. 

Suppose a stretched wire, fastened at 0, Fig. 7.1, is by some 
means bent in the form shown and then released. The tension 
in the wire will cause a return to its original position. But it can 
be shown mathematically that the wire can retain the form in the 

* The longitudinal vibration of a string is actually used in laboratory tests 
where a tone of high frequency is desired. The string may be rubbed by a sponge 
saturated with turpentine. The string is one-half wave-length long. From a 
knowledge of the wave-length and the velocity, the frequency can be computed. 

t The transmission of waves in solids involves, in general, another type o^ 
vibration also. 



same position given in Fig. 7.1 without any constraint whatever 
if the wire is moved to the left or right with a certain velocity v. 
In other words, a wave of this transverse displacement described 
will travel along the wire with a velocity v. From the very nature 
of the case one might expect the velocity in a perfectly flexible wire 
to depend only upon the tension of T, and the mass of the wire 
per unit length. That this is the case is stated in equation (7.1). 

Fig. 7.1 

If a transverse wave of displacement travels along the wire, 
it will be reflected at a fastened end and return in the opposite 
direction with the same velocity. The condition of stationary 
waves is fulfilled for there will then be two waves of equal ampli- 
tudes and frequency travelling in opposite directions. Since the 
wire is fixed in position at the ends, nodes will occur there. In 
Fig. 7.2 is what is called the fundamental or the lowest frequency 

Fig. 7.2 

of vibration. The vibration of next higher frequency with which 
the wire is capable of has one additional node as in Fig. 7.3. 
Successively higher frequencies that are possible have two, three, 
etc., additional nodes, respectively. The length of the wave in 
the wire is obtained in the same manner as is the wave-length in 
the stationary waves described in the Chapter VI. The distance 
between two adjacent nodes is one-half wave-length, but this is 
one-half of the wave-length in the wire. 



A string may vibrate simultaneously with the frequencies 
shown in Figs. 7.2 and 7.3. The lowest of these frequencies is 
called the "fundamental" and the others, respectively the first, 
second, etc., "overtones." Any one of these various frequencies 
is easily brought out by producing a corresponding node after 
the string has been bowed or plucked. 

Fig. 7.3 

It can be shown that the velocity of the transverse wave in 
the wire is 

= VJ> 


where T is the tension and m is the mass per unit length. In 
computations T is expressed in dynes * and m in grams per unit 
length. Let L be the length of the wire. If for the fundamental, 
L is equal to one-half of the wave-length, X, then, since frequency 
is velocity divided by wave-length (see equation 1.4), the fre- 
quency, », may be expressed as follows: 

v 1 [f 

This is the formula for the computation of the frequency of the 
fundamental of the string of any instrument. 

7.3. Measurement of Relative Intensities of Fundamental 
and Overtones. — The nature of the aerial wave given off by a 
* The weight of one gram is a force of about 980 dynes. 



string is of interest. From mechanical considerations it is real- 
ized that the vibrations of the string are communicated to the 
body of the instrument, such as the violin, and subsequently to 
the air. A problem is to measure the aerial vibrations, obtaining 
the relative intensities of the components. This is by no means 
easy for there is no direct method. If the sound falls upon a 
diaphragm and the motion of the diaphragm be recorded on a 
smoked paper, a curve will be obtained. This curve can sub- 
sequently be studied by an expert and the nature of the compo- 
nent vibrations obtained. But the diaphragm, because of its own 

resonance characteristics, will distort 
the relative values of the amplitudes 
of the components in the sound wave 
and, unless a correction can be made 
for such errors, the relative ampli- 
tudes of the sound waves cannot be 
obtained. Fig. 7.4 * shows such a 
mechanically produced tracing on 
the top line and below the analysis 
of it. There are three component 
vibrations having relative frequen- 
cies of 1, 2, and 3. In this particular 
case, the fundamental has the great- 
est amplitude, but this is not always 
true, either in the case of the violin, which was used in the present 
case or of other instruments. Yet the fundamental determines 
what is called the "pitch" 
of a musical sound. It is 
a curious fact in audition 
that the fundamental is 
more prominent to the 
ear than is the same intensity of any overtone. The character 
of the sound from an organ pipe is shown in Figs. 7.5 and 7.6. 
The former shows the resultant vibration produced by an organ 

♦Taken from Miller's "The Science of Musical Sounds," Macmillan, 1916, as 
are also Figs. 7.5 to 7.12. 

Fig. 7.4 

Fig. 7.5 


pipe and the latter the composition of it expressed in terms of the 
frequencies which have the ratios I : 2 : 3 : 4, etc., up to 12 times 
the fundamental. 

7.4. Instrumental Quality. — The difference between the 
quality of a tone of an organ pipe and that of a violin must lie 
in the relative intensities of the fundamental and overtones. This 
indicates that the overtones, al- 
though difficult to recognize by ear 
as such, are compositely very prom- 
inent in a musical tone. As a matter 
of fact the ear is a very sensitive de- 
tector of overtones and the develop- 
ment of the various types of musical 
instruments with their ensemble in 
an orchestra has depended upon just 
this sensitiveness. 

7.5. Sounds from Various Instru- 
ments. — In Figs. 7.7 to 7.12 are pre- 
sented charts showing the relative 
sound intensities of the components 
of various musical sounds. The ver- 
tical lines indicate the relative intensities. 

Fig. 7.6 

Along the horizontal 
the frequencies are distributed as on a piano, equal distances signi- 
fying an octave. These intensities have been secured from such 
curves as in Fig. 7.4 by making corrections for the resonating char- 
acteristics of the recording device and by transposing the facts 
concerning frequencies and displacement amplitudes into values 
of relative intensity. Relative sound intensities are always pro- 
portional to the square of the product of the frequency and the 
displacement amplitude. That is, if a tone of frequency 100 has 
an amplitude of 10 and a tone of frequency 500 has an amplitude 
of 2, the intensities would be equal. Hence, in obtaining relative 
intensities from amplitudes the frequencies must always be con- 
sidered. Fig. 7.7 shows the relative intensities in the sound from 


a tuning fork, the voice, the flute, the violin and the French horn, 
when each is sounded on the fundamental tone. Fig. 7.8 shows 
an analysis of flute tones. The lowest line shows the average 






ft 1 ■ m -•_ 









s i 











6 7 8 910 15 20 

Fig. 7.7. Distribution of energy in sounds from various sources 

composition of all the tones or the low register of the flute when 
played pianissimo. The second line from the bottom shows the 
middle register played pianissimo. When the lower register is 


I a * •— *- 

■ m • ■ 1 


§ P LOW 

5> — 




— T 1 1 1 1 1 1 — I I l I l l l 1 

1 2 3 4 5 6 7 8 9 10 15 

Fig. 7.8. Analyses of flute tones 

played forte, the octave becomes the most prominent as shown in 
the third line. When the middle register is played forte, the 
fourth line results. Figs. 7.9, 7.10, 7.1 1, 7.12 and 7.13 show Pro- 
fessor Miller's results for the violin, the clarinet, the oboe, the 




5 1 


3 1 











P 1 







5 6 


Fig. 7.9. Analyses of violin tones 







1 1 I I 1 l — I I I I 

2 3 4 S 6 7 8 910 


Fig. 7.10. Analyses of tones of the oboe and clarinet 




« « •-•■_- 









i ■ ■•■•»»! 



m I lllll 



! I I 1 jj I I 11 — 

5 6 7 8 910 15 

Fig. 7.1 i. Analyses of tones of the horn 





— T" 




65 129 259 517 1035 2069 4138 

Fig. 7.12. Analyses of tones of bass and soprano voices 



horn and bass and soprano voices. The variations in distribu- 
tion are wide. 

7.6. Recognition of Phase Differences of the Components. — 

Sound from a given instrument may vary in at least two respects, 
the relative intensities of the components and the relative phases 
of the components. It is a peculiar circumstance that the ear 

Fig. 7.13 

recognizes the combined tone as the same irrespective of the phase 
relations among the components.* But the trace of the resultant 
vibration differs widely with changing phase relations of the com- 
ponents. The reader can readily see this from the accompanying 
drawings, Fig. 7.13 and Fig. 7.14. In Fig. 7.13 are drawn two 

Fig. 7.14 

dotted curves representing the displacements in two simple har- 
monic vibrations (of a diaphragm, let us say), one having twice 
the frequency of the other. It is a note and its octave. The 
only difference between Fig. 7.13 and Fig. 7.14 is that the com- 
ponents are shifted in phase relative to one another. In each 
* Barton, "A Textbook of Sound," pp. 605-607. 


case one-half the sum of the two curves, or the resultant ampli- 
tude, is obtained by drawing the mean. The mean value curves 
are very different in the two cases. Indeed an inexperienced per- 
son would claim that the component vibrations could not be the 
same. Yet these two resultant vibrations give the same auditory 
sensation. This circumstance, that two dissimilar vibrations give 
the same sensation, makes the appearance of the resulting vibra- 
tion curve very deceiving. 


1. Of what two types of vibration is a string capable? Which 
one is used in music? 

2. Upon what factors does the velocity of a transverse wave in a 
perfectly flexible wire depend? 

3. What frequencies are possible in a string in terms of the funda- 

4. Why does the vibration of the string cause a vibration of the 
body of the violin ? 

5. Which component tone almost invariably determines the pitch 
of a sound ? 

6. Is there any difference in the perception of the fundamental and 
an overtone? 

7. Can phase relations of components be recognized? 

8. Upon what does the distinctive quality of sound of a musical 
instrument depend? 

9. Assuming that the velocity of a longitudinal wave in a steel 
string is 5,000 meters per second, how long must a string fastened at 
both ends be in order to give a frequency of 1,000 per second? 

10. What are the difficulties in detecting the components of a tone 
by eye inspection of the time trace of the vibration? 

11. In at least what two respects is there an opportunity of affect- 
ing the quality of voice by training? 

12. Why is the frequency of a string much higher with longitu- 
dinal than with transverse vibration? 

13. What difference would be noticed if one was suddenly enabled 
to detect relative phase changes in components? 

14. Why would the difference in quality of two violins depend 
upon the box, upon the nature of the string? 

15. Would the quality of tone of any instrument alter during the 
entire time of its rendition ? Answer by a consideration of one or two 



8.x. The Nature of Speech Sounds. — It is well to observe 
the physical mechanism producing speech, a vibrator, resonance 
chambers and an egress for the acoustic waves. During speech 
these three are almost constantly changing. If one speaks a syl- 
lable slowly so that the vowel sound therein may be said to be 
sustained, then there is an approximation to a steady physical 
state of affairs at the middle of the time interval for the syllable. 
But the vowel is not usually defined as limited to this midpoint 
at the interval. Moreover the component sounds of a vowel 
would not be the same in magnitude and frequency with different 
speeds of enunciation of the syllable. This is because of two 
effects. The establishment of resonance requires time. More- 
over if a resonant chamber receives an impulsive change of pres- 
sure it will give a rapidly decaying response in its own natural 
frequencies. For example, when the hands are clapped over the 
mouth of an empty jar, the sound heard is characteristic of the 
vessel itself. So it is easily realized that speech is a very com- 
plicated acoustic phenomenon, quite beyond the physicist's de- 
scription at the present time. For that reason the discussion in 
this chapter is practically limited to the composition of the sus- 
tained vowel sounds and even here the results are not at all com- 
plete. In the first instance the results presented will be those 
obtained by Professor D. C. Miller,* whose work represents the 
beginning of modern physical research on vowel sounds. 

8.2. The Vowels Used. — In an experimental test it is neces- 
sary to define the vowels tested. This was done by the following 
groups of words: 

* Figs. 8.1 to 8.6 are taken with permission from Miller's "Science of Musical 
Sounds." Published by Macmillan, 1916. 




























In considering the results obtained by Professor Miller, one 
must bear in mind that they have been secured through careful 
observation and the development of a technique in which all 
known corrections have been made. The intensities of the com- 
ponents in the first vowel as in father are shown in Fig. 8.1. 


Fig. 8.1. 


259 517 1035 




Loudness of the several components of the vowel in father 
intoned at two different pitches 

Here the vowel is sounded on two different pitches and in each 
there is greater intensity in the region of 922 cycles. The fre- 
quencies for which the measurements are made are given by the 
circles, the largest one indicating the fundamental tone. The 
smooth curves are drawn merely to indicate the general magnitude 
of intensity in a frequency region. They do not indicate actual 
intensity for any selected frequency. Fig. 8.2 shows additional 



experimental values of the components involved in this same 
vowel. B, C and D are the results of intoning at three different 
pitches. The curve A deserves special attention, because it is a 
composite of twelve such graphs. Twelve different notes were 
used as fundamentals and thus a distribution of computed values 

259 517 1035 



Fig. 8.2. 

Distribution of energy among the several partials of the 
vowel in father, intoned at various pitches 

were obtained. The reason that, with a given fundamental, the 
only frequencies indicated are integral multiples of the funda- 
mental, is explained in Section 15.6. Briefly it may be stated 
that if one has a vibration that is periodic, i.e., recurs at fixed 
intervals of time, its component frequencies must * all be integral 
multiples of the fundamental whose period is this interval of 

Hence in the analysis in any case only integral multiples of 
the frequencies are obtained. Returning to Fig. 8.2 it is easily 
seen that we have here a remarkable verification of the depend- 
ence primarily upon frequency regions. In Fig. 8.3 are shown 
the results with eight different voices sounding the vowel a in 

* The proof is entirely mathematical. 



« *_— 2 









> 5/7 1035 2069 



Fig. 8.3. Distribution of energy among the several partials of the 
vowel in father as intoned by eight different voices 

father on eight different fundamental tones. Fig. 8.4 shows 
the results with voices of the vowel in bee. Here the second 
resonance peak is drawn as a mean result of all eight voices. It 


259 517 1035 2069 




8.4. Distribution of energy among the several partials of the 
vowel bee, as intoned by eight different voices 



is evident that this is a vowel with two resonance regions; one 
about 300 and one about 3,000 cycles. This conclusion is verified 
by the use of an acoustic wave filter,* which will transmit the 








259 517 1035 2069 


Fig. 8.5. Characteristic curves for the distribution of the energy in 
vowels of Class I, having a single region of resonance 

lower and not the upper frequency region. If the vowel in bee 
is spoken through such a filter, it becomes almost the same as the 
fourth vowel in the list or as in gloom. The reason therefor is 















259 517 1035 2069 


Fig. 8.6. Characteristic curves for the distribution of the energy in 
vowels of Class II, having two regions of resonance 

readily seen in Fig. 8.5, wherein the latter vowel is shown to have 
a characteristic region in the neighborhood of 326 cycles or prac- 
tically the same as the lower one of the two regions in Fig. 8.4. 

Figs. 8.5 and 8.6 show two different types of vowels. In the 

* Acoustics, Stewart and Lindsay. 


former there is only one characteristic region and in the latter 
two characteristic regions. The vowel in ma is shown in both, 
for sometimes it shows two characteristic regions close together. 
This splendid group of curves shows with remarkable clearness 
the difference in the vowel characteristics. 

8.3. Characteristic Regions Are Resonance Regions. — In 

the discussion of resonance in a previous chapter attention was 
called to the resonance of the mouth, pharynx and larynx, and 
to the fact that the presence of this resonance increases the energy 
given off by the vocal chords in corresponding frequencies. It is 
to be expected, therefore, that the characteristic vowel frequen- 
cies would be produced by the resonance chambers. That this 
is true can be verified by adjusting the mouth and lips for the 
sounding of a certain vowel and then by presenting before the 
lips a vibrating tuning fork having a frequency in the character- 
istic region of that vowel. The tuning fork will be strongly re- 
enforced. The fact that the vowels can be whispered as well as 
spoken is also a verification of this point. Sir Richard Paget's * 
results show that there are always two simultaneous resonances, 
produced in two cavities separated by the tongue. Moreover he 
has by practice been able to demonstrate in the lecture room these 
two simultaneous regions, the source of sound being the clapping 
of hands in front of the mouth. 

An elaborate investigation of the characteristic regions has 
been made by Crandall f of the Bell Laboratories. The results 
are shown in the accompanying Figs. 8.7 and 8.8. In both of 
these figures the relative amplitudes at the different frequencies 
are plotted after taking into consideration the sensitiveness of the 
ear (see Chapter X). The "ordinates," vertical distances, in any 
one graph then represent what might be termed the relative 
"effective amplitudes" with that vowel. The ordinates for one 
graph are not to be compared with those of another since it is 
not the intention to show such a relation between the vowel 

* "Human Speech," Harcourt Brace & Co., London, 1930. 
t Bell Sys. Techn. Jl. y IV, 4, p. 586 (1925). 



sounds. The results of the investigations of Miller and Cramlall 
are in general agreement. The differences in methods of meas- 
urements are probably accountable for the differences in con- 














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elusions. Crandall's method of obtaining the amplitudes of the 
components involves the entire record of the vowel from start to 
finish. He finds the average characteristics throughout its dura- 
tion. There is first a period of rapid growth of 0.04 sec, second 
a middle period of about 0.165 sec, with variations but with 
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Fig. 8.8 

of a vowel sound. There is a variation with the individual and 
with the vowel, but the general relative magnitudes are retained. 
Crandall's results leave us with not so clear a distinction as to 
the number of resonance regions possessed by a vowel. It can- 
not be stated that any given vowel has only one or two resonance 
regions. Indeed any classification as to number of frequency regions 
would seem to involve crude limitations not fully justified. The 
differences and similarities of male and female vowel sounds are 


also to be noted. As a matter of fact, "resonance region" is an 
omnibus term and a description using it cannot be regarded as 
sufficiently specific. 

8.4. Clearness of Enunciation of Vowels. — All of the vowel 
sounds shown above have characteristic frequency regions that 
are in the upper half of the ordinary piano scale. Clear enuncia- 
tion thus does not need the bass quality of voice. Indeed, clear- 
ness must depend upon the relative amounts of energy that go 
into these characteristic frequencies. If all the energy is in a 
characteristic frequency region (or in all the characteristic regions 
as the case might be) the maximum clearness will be secured. It 
has often been observed that a child's enunciation of vowels is 
more clear than a man's and presumably the foregoing is a reason 

It is impossible to enunciate vowels clearly in singing unless 
the fundamental tone is lower than the characteristic frequency 
region of the vowel sounded. This is a well-known difficulty. 
Moreover, a great effort to sing the vowels clearly affects the 
quality of tone to an appreciable degree and prevents the artist 
from securing the best musical effect. 

Since vowel characteristics depend upon resonance and since 
time is required in getting the maximum intensity, it follows that 
vowel sounds must be sustained for a short period of time in 
order to be the most intense for a given effort on the part of a 
speaker. The sustained vowel sound probably enters into speech 
in other respects as well. But it is certain that rapidly spoken 
vowels are detrimental to clearness. 

In the discussion in this chapter the inference has been made 
that a steady-state vibration of the vocal chords can be secured. 
But practically this is not true. Indeed, some claim that the 
motion of the vocal chords is largely impulsive. One must expect 
that a detailed study of the mechanism will bring out many 
details at variance with the more simple picture in this chapter. 

8.5. Variation in Vowel Sounds. — Professor Miller's records 
were made of sustained vowels only, for the reason that the rec- 


ords of the vowels while they were being formed did not contain a 
sufficient number of like vibrations to make measurement pos- 
sible. The recent development of filters, both electric and acous- 
tic, has made the variable character of the vowels demonstrable. 
The following experiment by Stewart,* briefly referred to in Sec- 
tion 8.2, shows the variable character of the vowel e. The word 
eat was spoken through an acoustic wave filter which prevented 
the transmission of all frequencies above 2,900. But the mouth 
was formed for the vowel before it was sounded. Hence it had 
from its beginning the same adjustment as for a sustained vowel. 
A listener heard the word practically as oot. The removal of 
frequencies above 2,900 transforms the e into approximately 60. 
But if the word meet was spoken through the filter, it was easily 
recognized for there was a distinct sound of e. But if the vowel 
in meet was prolonged the e quality disappeared. In other words, 
the e heard consisted of frequencies below and near 2,900 which 
existed only temporarily while the vowel was being formed. The 
reason for the formation of lower frequencies can be understood 
by noticing the changes in the mouth during the utterance of the 
vowel. The mouth formed for an m must now be opened wider 
for an e. Assuming that the interior of the mouth is correctly 
formed for an e, the changing of the lips will increase the area of 
the orifice and raise the pitch of the resonance frequencies. The 
group of high frequencies should therefore alter during the for- 
mation of the vowel and in the direction demonstrated in the 
foregoing experiment. Inasmuch as the formation of vowels fol- 
lowing consonants requires a change in the shape of the mouth, 
there is in general a change in the components of any vowel when 
preceded by a consonant. Similar considerations will apply to a 
vowel succeeded by a consonant. In the case cited above the 
consonant / shuts off the vowel so quickly that no noticeable 
change in the vowel is produced. 

* Stewart, Phys. Review, 1923, Abstract of paper at Washington meeting of 
Am. Phys. Soc, April, 1923. 



1. Is the clearness of vowel sounds dependent upon the speed with 
which they are voiced and why? 

2. What reasons may be given for the differences in individual 
curves in Fig. 8.3? 

3. Why are the vowels sounded universally alike? 

4. In a previous chapter, the importance of the diameter of a 
sound reflector behind a speaker was emphasized. If the vowel sounds 
were the only ones of importance, what would you say about the 
minimum size of such a reflector? 

5. Why may a vowel sounded by a bass and a soprano voice be 
recognized as the same? 

6. Why does a highly pitched voice seem to carry relatively well 
in addressing a crowd? 

7. Discuss the possibility of a "monotone" learning to speak. 

8. According to the reasons given for vowel production, would it 
be possible to imitate speech by mechanical means only? 

9. What are the causes of unavoidable variations of a vowel sound, 
considering the different parts of its duration ? What additional vari- 
ations can be introduced? 



9.1. Energy Distribution. — One of the inquiries arising in 
speech concerns the relative amount of energy in the various fre- 
quencies used by a speaker. The recent development of ampli- 
fying tubes which can multiply electric power has enabled Messrs. 
Crandall and MacKenzie to measure this frequency distribution 
of energy in speech. These measurements have been made by 
using a 50-syllable sentence of connected speech and also a list 

004 p 

r *AL 












7000 2000 3000 4000 

1000 2000 3000 4000 

Fig. 9.1. Energy distribution 

of 50-disconnected syllables. The results show that the energy 
distribution varied more with individuals than with the nature of 
the test material chosen. Fig. 9.1 * shows in the first plot two 
composite curves. The one marked "male" is the mean of four 
curves representing the results taken with men. The curve 
marked "female" is the mean of two curves taken with women. 
* Physical Review, March 1922, p. 228. 



In the second plot is drawn the mean of all six curves. The ordi- 
nates of the curve are arbitrary and need not concern us here 
excepting that they are proportional to flow of energy. 

The test sentence was the first sentence of Lincoln's Gettys- 
burg address but with the addition of two words "quite" and 
"nice" to bring the total number of syllables to fifty and to 
"improve the balance between the vowel sounds." 







1 ! y 


f v 





2000 3000 



Fig. 9.2 

The sentence was uttered slowly syllable by syllable, in order 
to give time for the instruments to be read for each syllable. 
This method was not wholly satisfactory and will be subsequently 

In Fig. 9.2 are shown certain energy distributions. A was ob- 
tained by assuming Professor Miller's results for vowels; B was 
obtained experimentally by disconnected speech sounds, and C 
by connected speech sounds. 

The curves A and C do not agree. The difference should of 
course be contained in the consonants, but this explanation ap- 
pears to the above workers as not sufficient. It is evident that 
the difference may be explained qualitatively by the variable 



character of the vowels and the incorrectness of assuming that 
the sustained vowels give the correct energy distribution. 

These results, while giving the actual distribution of energy, 
do not give the relative importance of the different frequencies 
in clearness of speech, an important problem which is considered 
in the next section. 

9.2. Useful Energy Distribution. — Dr. Harvey Fletcher has 
presented the first report * on the ability of the ear to interpret 
speech sounds under different conditions of loudness and of dis- 
tortion caused by the elimination of groups of frequencies. This- 
is a direct method of determining the useful energy distribution. 
Inasmuch as the actual pronunciation of a vowel occurs in sylla- 


ErrecT upon interpretation or euminating 



















2000 3000 



Fig. 9.3 

bles, the tests were made with selected syllables. These were 
prepared in lists of 50, in which occurred a distribution of the 
three types, vowel-consonant, consonant-vowel, and consonant- 
vowel-consonant. In all 8,700 syllables were chosen. Fig. 9.3 
shows the results of articulation tests in graphical form. The 
abscissae, or the horizontal distances, represent the limit of the 
frequencies used and the ordinates the percentage of correct judg- 
ments as to the sounds received. In the curve having its begin- 
* Bell System Technical Journal, Vol. I, July 1922. 


ning at 100%, for the frequency of 1,000 the articulation is 86%. 
The interpretation is that if all frequencies below 1,000 were 
eliminated, 86% of the syllables could be correctly understood. 
Again, using the same curve, it is found that 40% of the syllables 
were understood if all the frequencies less than 1,950 were elimi- 
nated. The interpretation of the other curve is similar but the 
elimination of frequencies is above the limit instead of below. 
On this second curve 40% of the syllables were understood if all 
the frequencies above 1,000 were eliminated. From the two 
graphs, the following additional conclusions may be derived: (1) 
A system which eliminates all frequencies above 3,000 has as low 
a value of articulation as one which eliminates all frequencies be- 
low 1,000 cycles per second. (2) At a frequency of 1,550, or in 
the third octave above middle C, the importance of the frequen- 
cies higher than this value is as great as that of the frequencies 
lower than this value, the limit of consideration being 5,000. 

Several of the conclusions of Dr. Fletcher concerning the use- 
fulness of speech sounds are as follows: 

1. The short vowels, u, 0, and e are seen to have important 
characteristics carried by frequencies below 1,000. 

2. The fricative consonants s, z, and th are affected by elimi- 
nation above 5,000. 

3. The fricative consonants s and z are not affected by the 
elimination of frequencies below 1,500. 

4. The sounds th, f and v are the most difficult to hear and 
are responsible for 50% of the mistakes of interpretation. The 
characteristics of these are carried principally by the very high 

One of the practical applications of the information gained 
concerning speech sounds is in long distance telephony. In de- 
signing these telephone message circuits it is important that a 
frequency range be selected within which the transmission shall 
be made as excellent as possible. At the present time in America 
the range selected * is between 250 cycles and 2,750 cycles, with 
the endeavor to make the transmission for these limits attain a 

* Martin, Bell Sys. Tech. JL, IX, p. 483 (1930). 


certain standard. The loss in transmission at 250 and 2,750 
cycles is more than that 1,000 cycles, but by an amount that is 
not strikingly noticeable to the ear. The extension of the above 
stated range by a few hundred cycles evidently cannot, as shown 
in the foregoing figure, materially improve the understanding of 
telephone messages. But if, in the future, much more perfect 
station instruments are produced, more naturalness will be at- 
tained and largely because of the better reproduction of the 
fricative constants. 

9.3. Speech Energy. — Crandall and MacKenzie,* using fifty 
syllables spoken in a normally modulated voice by six different 
speakers (four men and two women) give 125 ergs per second as 
the average acoustical output. Sabine t using a very different 
method obtains for certain vowels an output varying from 271 
to: 70 ergs per second. An acoustical output of 100 ergs per sees, 
ond would be 10 millionths of a watt or 10 microwatts. The non- 
technical reader can compare this value with the lamp at his 
study table, rated at 40 watts. The actual power output jn 
speech is approximately two ten-millionths of that of a 40 watt 

Interesting experiments have been performed to ascertain the 
relative amount of flow of energy or power occurring in conver- 
sational speech. Sacia and Beck t have presented their results 
in the form shown in the accompanying Table IV. 

The power during the existence of a given sound is averaged 
and this is called the "mean power." The "peak power" is the 
average of the maximum value for two speakers. As may be 
seen the vowels rank the highest, the semi-vowels next and the 
consonants the lowest. 

* Crandall and MacKenzie, Phys. Reo. t 19, 1922, p. 221. 
t P. E. Sabine, Phys. Rev., 22, 1923, p. 303. 
% Bell System Tech. JL, V (1926), p. 393. 


Table IV 



Relative Power, 

Arbitrary Units 

Mean Power 


Peak Power 


Normal Values 

Values for 16 

for 2 


































































































6 I 
















J 5 


! get 














1 1 

* The dash indicates that observations were not available. 


1. Show from a consideration of Figs. 9.1 and 9.2 that a man's 
clearness of enunciation of vowel is less than a woman's. 

2. What other factors are involved in the production of superior 
clearness of speech? 

3. What is a striking illustration of the fact that clearness of 
speech may depend upon very small sound intensities? 



io.i. Energy Required for Minimum Audibility. — There 
have been many observations made of the sensitivity of the ear 
and a summary of results and methods would be impracticable. 
The results of Wien,* Kranz t and Fletcher and Wegel t are the 
most reliable, though they differ by much more than can be 
accounted for in the difference of the observers. The number of 
ears used in these investigations were, respectively, 3, 14, and 72. 
If the results are weighted by these numbers and the means taken, 
the best values of minimum audible pressure, expressed in dynes § 
per sq. cm., are found to be: 

Threshold pressure 








These values show that the threshold value of pressure (or the 
minimum audible pressure) is fairly constant above 1,000 vibra- 
tions per second and increases rapidly at lower frequencies. At 
an octave above middle C the pressure is approximately .001 
dynes per sq. cm. This is a region of important frequencies and 
may be regarded as the most representative single value of the 
sensibility of the ear. From these values and a knowledge of the 
plane waves, it can be shown that the minimum rate of energy 
flow through a square centimeter in such a wave is of the order 

* Wien, Arch./, ges. Physiol, 97, p. 1, 1903. 

t Kranz, Phys. Rev., 21, p. 573, 1923. 

% Fletcher and Wegel, Phys. Rev., 19, p. 553, 1922. 

§ In expressing the dynes per sq. cm. of a pressure that varies periodically, 
positive, zero, negative, etc., one cannot give the average pressure for that would 
be zero. But since the intensity of the sound varies with the mean square of the 
pressure, it is customary to refer to the pressure of a sound wave as the square 
root of this mean square. This gives a value proportional to the square root of 
the intensity and proves convenient. The maximum pressure can be obtained by 
multiplying this value by yjT. 




4 X io~ 16 (or 4 divided by the number ten raised to the sixteenth 
power) watts or 4 X io~ 10 microwatts.* 

10.2. Limits of Audibility. — The limits of audibility can be 
best expressed in the form of graphs. Fig. 10.1 is taken from the 
work of Fletcher and Wegel.f The full-line portion of the lower 
curve is a plot of the observations of minimum audibility men- 
tioned in the foregoing paragraph. The ordinates represent the 
square root of the mean square of the pressure from the normal. 
The units at the right are explained in Section 10.6. 

32 64 126 256 612 1024 2046 4096 6192 16364 FREQUENCY DV 

Fig. i o.i. Any sound that can be heard lies within the field 
outlined here. Areas covered by the most prominent speech sounds 
are indicated. 

The upper curve represents experiments on 48 normal ears 
and is regarded as the " maximum" because at these pressures 
there is a sensation of feeling which limits the pressure used in 
any hearing device. This statement assumes that the sensation 
of feeling is approximately the same in abnormal ears, and hence 
it can be only an approximation, which varies with the nature of 
the abnormality. It is interesting to learn that the intensity for 
feeling is about equal to that required to excite the tactile nerves 
in the finger tips. 

* The square root of the mean square of pressure in dynes can be shown to be 
equa l to V20.5 X E> w here E is the flow of energy in ergs per sq. cm. per sec. and 
to V20.5 X io 7 X E> where E is the flow of energy in watts per sq. cm., £ being 
the flow of energy produced by a plane wave having this pressure. 

t See Wegel, Bell System Technical Journal^ Vol. I, No. 2, 1922. The form 
used in the figure is that suggested by J. C. Steinberg of the same laboratory. 


The full-line curves have been extended as dotted curves and 
made to meet at approximate values of the frequency limits of 
audibility. The lower value, 20 vibrations, is really a high value 
selected because there is not an agreement as to the smallest fre- 
quency that can be recognized as a tone. The upper limit, here 
represented as 20,000, varies distinctly with age, being higher for 
the young and becoming lower with age. It has not been demon- 
strated whether this change is pathological or physiological. The 
cross hatched areas show the ranges of pressures and frequencies 
for the sounds indicated. The important frequencies in voice 
timbre and sibilance are also shown. 

10.3. Deafness Defined in Dynamical Units. — These authors 
have found that in the region up to 4,000, persons of normal hear- 
ing require a pressure variation of approximately 1/1,000 dynes 
per sq. cm. for audibility. Persons called "slightly deaf" require 
a pressure of 1/10 dynes per sq. cm. A person requiring 1 dyne 
per sq. cm. can usually follow ordinary conversation. Those who 
require 10 dynes per sq. cm. need artificial aids to hearing. 

In this connection it is also interesting to note that even if 
sounds could be amplified indefinitely, yet there is a limit to 
amplification because one cannot use an intensity great enough 
to cause pain. Hence only the deaf that have a range of pressure 
sensitivity between 1,000 and 1/1,000 dynes can be made to hear 
through artificial aids such as amplifying tubes and microphones. 
The above comments do not consider the totally deaf who by 
definition cannot be made to hear. Also, the conclusions as to 
painful effect have been obtained by a study of those not deaf 
and hence the numerical limits here fixed are open to question.* 

10.4. Types of Deafness. — That deafness may have many 

causes has been well known to otologists but only in recent years 

have attempts been made to study types of deafness by actual 

* Reger, of the Psychological Laboratory at the University of Iowa, reports 
that the congenitally totally deaf as well as those who have lost their hearing 
and vestibular reaction due to cerebral meningitis, experience the sensation of 
feeling at approximately the same sound pressure at various frequencies as do 
individuals with normal hearing. He believes that the sensation arises from the 
stimulation of pain receptors within the tympanic membrane. 





I ! Is . . S I 1 I 





















' m y 



*5 o © 

^ - s § § § 



5 « 


Co * 




I 8 








'_ . 


t S * S S § § 



measurement of hearing. In Fig. 10.2 are represented * four 
curves of ear tests, plotted as in the original data. As will be 
readily seen the four cases vary widely. The ordinates on the 
right represent the relative loudness change as observed by a nor- 
mal ear. The upper and lower curves in each case correspond to 
the similar curves in Fig. 10.1 . A change of 10 in loudness means 
that there is an apparent change of 10 times the loudness. 

10.5. Loudness. — One of the most interesting contributions 
to this field is that of Professor Wallace C. Sabine.t He used 
several different musical instruments, obtaining an equality of 
loudness for the seven frequencies shown in the accompanying 
Table V. In the second column of this table is shown an arbi- 
trary number obtained for each frequency at equal loudness by 
dividing the actual intensity by the minimum intensity required 
for audibility of that frequency. This column therefore repre- 
sents in each case the actual intensity in terms of the minimum 
audible intensity for that frequency. For a frequency of 64 the 
intensity is 0.7 X io 6 times the minimum audible intensity for 
that frequency. 

Table V 


Intensity Ratio 


0.7 X io 6 












1 .048 " 

The application of these comparisons is perhaps not at once 
evident. Assume that one is listening to an airplane, flying first 
near by and then at a distance. Assume that when it is in the 
near position a number of frequencies are heard, apparently of 

♦These are from curves shown at Philadelphia by Dr. Harvey Fletcher, in 
1923. See "Speech and Hearing," D. Van Nostrand Co., p. 200. 
t Collected papers in "Acoustics," p. 129. 



equal loudness. What will occur when the airplane recedes to a 
distance, assuming, for the moment, that the intensity of each 
frequency will decrease inversely as the square or any other power 
of the distance? The intensity of each frequency will be cut 
down to the same fraction of its original intensity. If this frac- 
tion is small enough some of the frequencies will be thereby cut 
down below audible intensity and cannot be heard. Hence the 
quality of sound from the airplane will change. Suppose the 
fraction is 1/100,000. Divide each number in the second column 
by 100,000 and it is seen at once the intensity of the 64 and the 
4,096 frequencies have fallen below the intensity of minimum 
audibility in each case. If the fraction is 1/10,000,000, it is easy 
to see that the 128, 256 and 2,048 cannot be heard. As the frac- 
tion gets less and less, additional frequencies will disappear. The 
last frequencies to be heard will be in the region of 1,024. We 
have here, then, a clear explanation of the change of the quality 
of sound with distance only. As hereinbefore stated, diffraction 
and reflection are not discussed. They have independent influ- 
ences in the variation of quality with distance. Steinberg * has 
shown that if a component tone has less than its own threshold 
pressure, it will not add to the loudness of the complex tone. 
Thus, in harmony with Sabine's results just stated, a complex 
sound can be heard no further than its most persistent component 
can be heard alone. It is obvious that if the entire energy avail- 
able be expended in a tone of just one frequency, it can be heard 
further than the same energy distributed in any combination of 
frequencies whatever. This accounts for the apparently different 
carrying qualities of sounds from various sources. These con- 
siderations are, however, not the only ones applicable when other 
masking sounds are present. 

10.6. Weber's Law — Fechner's Law — Sensation Units. — 

There is a general law in psychological literature that evidently 
is a law of the nervous system and is independent of the nature 
of peripheral organs. It may be stated as follows. The least 
* Steinberg, Physical Review y 2, 26, p. 507 (1925). 


detectable change in any stimulus is proportional to the intensity 
of that stimulus. This law of Weber has been verified in many 
directions, such as hearing, vision and feeling. In acoustics we 
are interested to know the relation between intensity and loud- 
ness. Although not an exact statement, a reasonably accurate 
guide has been found in assuming the sensation of loudness 
to be proportional to the logarithm * of the intensity in the 

This logarithmic law is usually known as Fechner's law and is 
expressed as follows: 

S — c logio^> + a. 

Here S is the sensation or loudness, p is the pressure and a and 
c are constants for any one frequency. For frequencies varied 
from 100 to 4,000, c changes not more than 10%. MacKenzie t 
has performed experiments on the relative sensitivity of the ear 
at different levels of loudness and his results confirm the correct- 
ness of Fechner's law. He found that c is a constant. Now 
loudness should be measured in units so that twice the sensation 
of loudness would be indicated by twice as many units. It is 
observed in the preceding equation that, since the sensation of 
loudness varies with the logarithm of pressure, one cannot use 
for units of this sensation, units of pressure. A new unit is desir- 
able and the literature is gradually adopting as a definition of 
sensation units ', S = 20 logio p. The value of S is said to be in 
"decibels." The sensation level, "SL," of a tone is defined by 

SL = 2ologiop//>o, 

where po is the threshold pressure. The sensation level is really 
the number of sensation units in decibels that are required to re- 
duce the tone to the threshold limit. A difference of one decibel 

* The logarithm is a convenient measure of a number. The number 10 raised 
to the fourth power is 10,000. The logarithm of 10,000 is then 4, if the logarithm 
be written, logio io,ooo. Also logio 10 = 1, logio 100 = 2, logio 1000 = 3. Like- 
wise every number has a logarithm. It is noticed that the logarithm changes very 
slowly as compared to the number itself. 

t MacKenzie, Phys. Rev., 20, 1922, p. 331. 



in a sensation level may be produced by approximately twice the 
minimum perceptible difference in intensity described in the next 
section. The range of speech sounds is usually from 40 to 60 
decibels, an average whisper four feet away is 20 decibels, a rustle 
of leaves in a gentle breeze is 10 decibels, and very loud sound, 
almost painful, is about 100 decibels, in each case the number of 
decibels referring to the number required to reduce the tone to 
the threshold limit. 

10.7. Minim um Perceptible Difference in Intensity. — The 
most comprehensive results of minimum perceptible difference 
have been presented by Knudsen * and may be expressed as 


















0.1 1 







Here SL is the sensational level or the number of units as above 
defined required to reduce the tone to the threshold, AE is the 
minimum audible increment in intensity, and E is the intensity. 

If Weber's law above stated were true, -g- would be a constant. 

Wien in 1888 first showed that this could not be the case. Knud- 
sen now shows that at the same loudness or sensation level, the 
ratio is practically independent of frequency, varying only 10% 
with frequencies of 100 to 3,200 cycles. Macdonald and Allen f 
have recently published data showing a similar variation of 

A£ ... 
-=r with E. 

10.8. Audibility of a Tone Affected by a Second Tone : Mask- 
ing Effect. — The experiments in this subject at the Bell Tele- 
phone Laboratories % have been very extensive and have an im- 

* Knudsen, Phys. Rev., 21, p. 84, 1923. 

t Macdonald and Allen, Phil, May 9, p. 827, 1930. 

% Wegel and Lane, Phys. Rev., 23, p. 266, 1924. 



portant bearing upon the theory of hearing. The phenomenon 
can be appreciated through a brief study of the results obtained 
by using a constantly sounding tone of 1,200 cycles and observing 
its masking effect upon other frequencies, the same ear being 
used. A definite arbitrary measure for " masking " must be given 
in order to express the results quantitatively. The investigators, 
Wegel and Lane, have selected the following: If without the 
masking tone, the threshold pressure of a given frequency is pi, 
and with the masking tone it is p iy then the measure of masking 

is — • In the accompanying Fig. 10.3 the three curves refer to 



800 1200 2000 


3000 4000 

Fig. 10.3 

the results with three intensities of the tone having a frequency 
of 1,200; namely, 160, 1,000, and 10,000 times its minimum 


audible intensity. The ordinates are *j- as already described. 

Several points are to be noted: 

1. There is a decided masking which increases as the fre- 
quency of the masking tone is approached. 

2. The masking effect increases with the intensity of the mask- 
ing tone. 

3. At higher intensities the masking occurs as if the additional 
tones of 2,400 and 3,600 were present. This indicates that those 
tones have been created by the ear itself, and, as will be explained 
at a later point, this is actually the case. 



The authors just quoted have utilized the study of masking 
in a consideration of the frequency regions on the basilar mem- 
brane and their conclusions may be represented in the accom- 
panying drawing, Fig. 10.4. As it is not the purpose of this text 
to discuss the anatomy of the ear, no explanation will be offered 
other than in the figure itself. Those who are not familiar with 
the ear are referred to the original article. The results in Fig. 

Oval ^^ 

Window 5 w 



Fig. 10.4. 

Distance from oval window (Millimetres) 
Characteristic frequency regions on basilar membrane 

10.4 should not be regarded as final but only as representing a 
stage in the progress of the establishment of a correct theory of 
hearing. It may be well to remark that there is at present no 
fully accepted theory of hearing. Several recent contributions 
to the discussion are contained in the account of a symposium of 
the American Acoustical Society in 1929. 

10.9. Hearing in the Presence of Noise. — It has been re- 
peatedly noticed that many people appear to hear better in the 
presence of a noise. The term used to describe this phenomenon 
is "paracusis." Knudsen and Jones * have removed much of the 
mystery in connection with the apparent better hearing of the 
"paracusic" in the presence of a noise. They find that, strictly 
speaking, the acuity of hearing is decreased by the presence of a 
noise for all persons of normal hearing and for all the partially 
deaf. Further, the individuals with impaired hearing of the per- 
ceptive t type do not hear as well in the presence of a noise. 

* Knudsen and Jones, "The Laryngoscope," $6, p. 623, 1926. 

t The terms "perceptive" and "conductive" are taken from the original paper. 
The former evidently refers to the usual means of hearing and the latter to the type 
where the vibrations reach the organ of hearing by bone conduction. 


Paracusis is found only in the individual having hearing of a con- 
ductive type and usually one having a marked bilateral lesion, 
wherein the difficulty is the transmission through the bones of the 
ear. And in these cases the phenomenon occurs not because of 
increased acuity, but rather because of certain other advantages 
inherent in his impairments. For example, his partial deafness 
is chiefly in low tones, whereas his acuity is good in the region of 
most importance to speech, 1,000 to 2,000 cycles. In the pres- 
ence of a low pitched noise, such an individual would enjoy a 
relative advantage in listening to conversation, when compared 
with the person having normal hearing. For the energy of noise 
is usually well below 512 cycles. Another advantage rests in the 
fact the lower tones are more important in the masking effect 
than the high ones. Hence, in the case of a noise covering a large 
range of frequencies, the elimination of the low tones by the indi- 
vidual mentioned would result in a noticeable relative advantage 
in the presence of such a noise. 

10.10. Minimum Time for Tone Perception. — The data 
available on the minimum time for tone perception are discordant. 
Fletcher * states that at 128, 384, and 512 cycles the values of the 
time required for the perception of weak tones are those corre- 
sponding for 12.1, 24.1, and 29.6 cycles, respectively. These 
correspond to the time intervals, .0946, .0627, and .0579 seconds. 
For tones of medium strength, the time is noticeably reduced. 
But the phenomenon is more significant than appears upon the 
surface. The minimum time for tone perception may not have 
even the same meaning for different individuals. A tone is rec- 
ognized as such only when it is identified as within a certain fre* 
quency range. With one individual this range may be larger than 
it is with another.f 

10.11. Minimum Perceptible Difference in Frequency.-— 
For frequencies between 500 and 4,000 the least perceptible dif- 
ference in frequency is about 0.3 of one per cent of the frequency, 

* "Speech and Hearing" (D. Van Nostrand Co.), p. 153. 

t See Stewart, Journal oj the Acoustical Society of America, II, 3, p. 325, 1931. 


f, and is fairly constant. Calling this difference A/", then this 


percentage is -j — 0.003. At 2 ^°» I2 °> an d 7°> tne values are 

0.004, 0.006, and 0.009 respectively. These are reported by 
Knudsen.* The method was to sound one tone at a time. Just 
to what extent the least perceptible difference depends upon the 
length of time the tones are heard is not known. It is fairly safe 
to opine that this time duration is a factor which should be con- 

10.12. The Vibrato. — The interest of this paragraph is the 
audition of the vibrato rather than a report of present detailed 
knowledge concerning the vibrato itself. Vibrato and tremulo re- 
fer to effects which can be produced by the voice and by some 
musical instruments. The former appears to the ear to be a fluc- 
tuation of intensity of about 5.5 to 8.5 times per second. It 
occurs in the natural singing voice and Metfessel t states that 
the vibrato is a cycle of frequency variation with an average rate 
of seven cycles per second and an average extent of a musical 
half-tone. The ranges of variation of the number of alterations 
of frequency is from 5.5 to 8.5 per second. The range of the 
variation of the frequency of the tone is from a tenth to a whole 
tone. These results were obtained by tests made with the voices 
of over forty artists. A greater variation of frequency and in- 
tensity than found with artistic vibratos may be called a tremulo. 
Now one of the interesting points concerning the vibrato is that 
although the frequency alteration is well within the ability of the 
ear to detect under more favorable circumstances, yet here, with 
the rapid alteration of about 14 times per second, the ear cannot 
detect a change in frequency.! Evidently this is in part caused 
by the brevity of the time duration of the tones. Another curi- 
ous result is that the change in intensity seems to be marked, 
indeed much more so than accountable by a direct intensity 

* Physical Review, XXI, 1 (1923). 

t M. Metfessel, Abstract, Acoustical Society of America, meeting December 13, 

% Tiffin, Joseph, Psychological Monographs , 193 1, 1 4. 



effect. Whether or not this is explained in the nature of the ear 
response is not known. 

The tremulo is similar to the vibrato except that the variation 
in frequency is large enough to be heard. Apparently it presents 
no new problem of the kind just described. 

10.13. Loudness of Complex Sounds. — A study has been 
made of the absolute loudness of a complex sound and Steinberg * 
finds that the loudness can be expressed in the form of an equa- 
tion the factors in which have been determined. 

10.14. Combination Tones. — There are two chief combina- 
tional tones, called "summation" and "difference" tones. If 
two tones, having frequencies of ti\ and »2, are sounded, the sum- 
mation tone has a frequency («i + » 2 ), and the difference tone a 
frequency (#1 — n 2 ). If, on a piano, middle C or C 3 (see Section 
14.4) and Fz are struck vigorously, the tone F x can be heard. 
This is a difference tone. If the keys Fj and C 2 are struck, the 
summation tone is that practically of A. The reader is referred 
to Barton t for a detailed description of the best method of bring- 
ing these tones into evidence. 

10.15. Frequencies Introduced by Asymmetry. — If a vibrat- 
ing body is displaced, it exerts a restoring force. It may be said 
that there is symmetry if the magnitude of this force is independ- 
ent of the direction of the displacement and dependent only upon 
the amount. If there is asymmetry this is not the case, and the 
restoring force will change in magnitude if the displacement 
changes in sign only. It can be shown % that in such a vibrator 
the fundamental will always be accompanied by all the overtones 
which are integral multiples of the fundamental. Also, mathe- 
matical analysis shows that if the restoring force is made pro- 
portional to the square of the displacement, and thus asymmet- 
rical, the response will not consist merely of vibrations corre- 

* Steinberg, Phys. Rev., 25, 1925, p. 253. 
t Barton, loc. cit. y articles 297 to 301. 

X Barton, loc. cit. There are many asymmetrical vibrators. With this one, 
the restoring force depends upon both the amplitude and its square. 


sponding in frequencies to those of the impressed forces. If this 
vibrator is set in motion by two forces having frequencies of p 
and q y there result vibrations * having the following frequencies: 
py $9 ^P *t" ?)> (P ~ ?)> 2 P an( * 2 $* But tne amplitudes of the 
four additional tones are of peculiar interest, for they increase 
rapidly with increasing amplitudes of the primary vibrations p 
and q. The bearing of this conclusion of the assumed case upon 
the hearing of combinational tones is quickly understood. The 
drumskin of the ear is an asymmetrical vibrator and it is not sur- 
prising that, in case frequencies p and q are sounded with suffi- 
cient intensity, the summation tone (p + q) and difference tone 

(p ~ 2) w iN be heard. 

The condition of symmetry is not sufficient to cause a vibrator 
to give only that frequency which is impressed upon it. In addi- 
tion, the restoring force must be proportional to the displacement. 
If the restoring force depends not only upon the first power, but 
also upon the cube of the displacement, then the vibrator, with 
an impressed force of frequency^ will introduce the frequency 3^. 

10.16. The Ear an Asymmetrical Vibrator. — It has been pre- 
viously stated that if two tones of different frequencies actuate 
an asymmetrical vibrator, there will be produced by the latter 
the summation and difference tones. This fact causes the ear 
sometimes to misjudge the frequencies present in a complex tone. 
Dr. Harvey Fletcher t has studied the criterion for determining 
the pitch of a musical tone. He found that if the frequencies, 
100, 200, 300, etc., up to 1,000, each having the same pressure 
amplitude, were sounded together and then again with the 100 
removed, no difference could be detected by the ear. Even if the 
elimination continued leaving only the 800, 900 and 1,000, the 
pitch seemed to correspond with 100 vibrations per second. In 
fact any three consecutive components gave this same pitch. If 
the components aoo, 400, 600, 800 and 1,000 were used, the pitch 
corresponded to 200 cycles. Any consecutive three of these tones 
gave the tone of 200, but weakly. 

* The statement as to frequencies represents an approximate solution, 
f Fletcher, Physical Review, 23, 1924, p. 427. 


One might conclude that the pitch of a musical tone is always 
determined by the common difference in the frequencies of the 
harmonics, but this is not correct. The combination 100, 300, 
500, 700, 900, did not give a pitch of 200 but sounded more like 
a noise, but with the frequency 200 distinctly audible. The fact 
that components are multiples of a common difference seems to 
give the pitch with certainty. 

These illustrations of the asymmetrical character of the ear 
lead to the conclusion that the quality of a complex sound as 
judged by the ear must depend not only upon the relative ampli- 
tudes of the components but also the actual amplitudes. For 
the asymmetrical character of the ear is brought into greater 
relative prominence by greater amplitudes of vibration. Hence 
if a complex sound is made fainter, but with the relative intensi- 
ties of the components constant, the ear would notice a difference 
in quality. As the sound is made louder, the lower tones would 
become increasingly prominent. In this manner a loud-speaker 
which may give a faithful reproduction would, if sufficiently loud, 
appear to over-emphasize the lower tones. The asymmetrical 
nature of the ear introduces a difficult problem into the construc- 
tion of acoustical instruments. 

10.17. Use of Combinational Tones in the Organ. — The use 
of a combinational tone in a musical instrument actually occurs 
in the pipe organ. An open pipe 16 feet in length will give ap- 
proximately 32 vibrations per second. But to give 16 vibrations 
per second a pipe 32 feet in length would be required. This is 
usually too long to install in the recess set apart for the organ. 
The tone is obtained by the combined use of two pipes of 16 ft. 
and iof ft. acoustic length, giving 32 and 48 cycles. Thus 
there is obtained a combinational tone of 16 cycles. Apparently 
such practice is common. 

But there is no reason why an organ pipe should be straight. 
The reflection at a bend in a pipe is caused by the fact that 
the successive wave fronts do not remain parallel and there is a 
certain amount of interference which depends upon wave-length. 


The longer the wave-length, the less the interference and conse- 
quently the less the reflection at a bend in a pipe. Also, the 
larger the diameter of the pipe, the greater the reflection if the 
diameter of the pipe is small in comparison with the wave-length. 
This is shown in Fig. 12.4. It is possible, then, to have a pipe 
double back upon itself and thus reduce the length it occupies. 
This construction is now also utilized in organ building. 

10.18. Pressure of Sound Waves. — Lord Rayleigh has shown 
that a plane wave of sound striking a wall perpendicularly will 
exert a constant excess of pressure determined by the formula, 

P = *E, 

wherein p is the excess pressure and E is the energy per unit 
volume in the incident wave. This phenomenon has a bearing 
upon the preceding discussion, for the amplitude of the resulting 
waves when two frequencies of n\ and #2 are sounded fluctuates 
with a frequency (»i — « 2 ). E in such a wave would fluctuate 
with this frequency and hence p also. Such a fluctuating pres- 
sure would cause any vibrator to respond with the difference fre- 
quency (ni — W2). But this pressure proves to be very small. 
Computation shows that lE is about one millionth the value of 
the pressure of the incident wave, if the latter is one dyne. For 
less pressures in the incident wave the fraction is even less. Thus 
it is learned that although the difference tone (»i — n*) does have 
an objective existence in the air, yet the ear cannot hear it. It 
is usually referred to as a subjective tone. The hearing of the 
difference tone really depends upon the asymmetrical vibration 
of the ear. 

10.19. Intermittent Tones. — If a frequency of n vibrations 
per second is interrupted u times per second, and if u is smaller 
than 77, a frequency of u per second will be heard. If the ampli- 
tude of the n vibrations is varied u times per second, again the 
tone u is heard. F. A. Schultze * considers the case theoretically 

* Schultze, Annul d. Physik, 26, 7, 1908, p. 217, and Science Abstracts, No. 1657, 


and shows that there are objective tones present having the fol- 
lowing frequencies:/), (p - «), (p + «), (p ~ 2«) and (p + iu). 
From these may be formed the combinational tones u, lu, 3a, 
etc., and {ip + iu) y (ip + 2«), etc. It is obvious that the fre- 
quency u may exist on account of the changes in mean pressure 
that occur with that frequency and on account of the asymmetry 
of the ear. 

10.20. Intensity and Pitch of a Blend of Sounds. — Two tones 
of exactly the same frequency and the same phase at a point will 
give there an intensity four times that of one alone, for the am- 
plitudes add and the intensity is proportional to the square of 
the amplitude. But suppose there are n such tones, then the 
intensity would be proportional to w 2 . But only with extraor- 
dinary care could such a result be achieved, n violins playing 
in an orchestra must give phases at random at the auditor's ear, 
even if the frequencies are identical. The late Lord Rayleigh * 
has shown that in such a case the intensity is not proportional 
to n 2 but to n. It can also be shown that if there are a number 
of tones of nearly the same frequency and of approximately the 
same amplitudes, the resulting tone will be judged by the ear as 
having a pitch which is an approximate mean of the two extremes. 
The hum of a swarm of bees is an illustration. 


1. What physical phenomenon may sometimes lead to an incorrect 
impression of the pitch of the lowest tone present? 

2. What frequencies will an asymmetrical vibrator introduce when 
a pure tone is sounded? 

3. In what case will a sound wave exert an excess of pressure? 

4. What personal experiences can be explained by Sabine's work 
on loudness? 

5. Explain why one tone of a frequency will "carry" farther than 
a complex tone of the same total acoustic flow of energy. 

6. The frequency stated for least audible pressure, 2,048 cycles, as 
indicated in Fig. 10. 1 is not in numerical agreement with the second 
column of Table V. Show that there is not necessarily a discrep- 

* Rayleigh, Scientific Papers> Vol. 3, p. 52. 



ii.i. Binaural Intensity Effect. — There is no doubt but that 
difference of intensities at the ears is a factor, though small, in 
the auditor's location of a source of sound. It is the purpose of 
this section to state what sense of direction of the source is given 
by intensities at the two ears. Stewart and Hovda * discovered 
that there was a very precise mathematical relation between the 
ratio of the intensities at the ears and the apparent direction of the 
source. If one draws an imaginary plane midway between the 
ears and the perpendicular to the line joining them, then the 
apparent position of the source in a horizontal plane may be de- 
scribed by the angle its direction makes with this median plane. 
If this angle be denoted by and the intensities at the ear by I\ 
and J 2 > then the technical mathematical statement is 

flcclog^r- (ii.i) 

This states that the angle, 0, between the median plane and the 
direction of the sound is proportional to the logarithm f of the 
ratio of the intensities at the ears. If ii, the intensity at the 
right ear, is the greater, then is measured to the auditor's right 
of the median plane. If h is the greater, the angle is to the left. 

In obtaining the above results the apparatus was arranged so 
that the sound involved was a pure tone emitted by a tuning fork 
and there was no difference of phase at the ears. 

From the data not here reproduced the following conclusions 
may be derived: 

* Stewart and Hovda, Psych. Review, XXV, No. 3, 191 8, p. 242. 

fSee reference in Section 10.6. In (11.1), the form, logio, could be used, 
but the equation would remain true if another number than 10 were used. So 
it is left without such a number indicated. 




( 1 ) Formula ( 1 1 . 1 ) was found correct for several observers and 
for frequencies of 256, 512, and 1,024. 

(2) The binaural intensity effect does not account for a hear- 
er's ability to locate a source of sound for to produce a certain 
apparent in the experiment, a much greater ratio of intensity is 
required than can exist in an actual case with the head casting 
an intensity shadow for the sound from a distant source. 

Later experiments * were performed with 16 observers and 
with frequencies from 200 to 2,000. Four of the observers were 
tested at frequencies up to 4,000. These experiments were per- 
formed in such a manner that they did not show the accuracy of 
equation (11.1), but they did clearly indicate the frequencies at 
which the intensity effect did not exist at all. Some of the con- 
clusions of these experiments are: 

(1) With ten of the sixteen observers, the binaural intensity 
effect ceased to exist throughout one or more bands of frequencies. 
(When it ceased to exist, the source of sound appeared to remain 
directly in front of the observer, though the intensity ratio 
changed within wide limits.) 

(2) These frequency bands all occurred above 800 cycles and 
were of different widths. 

(3) In certain frequency regions there appeared to be two 
sources of sound, one stationary in front and one moving about 
with changes in intensity ratio. 

Table VI 


Lapse range 


Lapse range 






1 1 50-1 250 




























* Stewart, Phys. Rev., Vol. XV, No. 5, 1920, p. 432. 


(4) There is a wide variation among individuals in regard to 
the above. 

Table VI shows the results so far as the absence of the 
intensity effect is concerned. 

1 1.2. Binaural Phase Effect. — By " binaural difference of 
phase effect" is meant the alteration of the angular displacement 
from the median plane of the apparent source of the fused sound 
when varying differences of phase of a given frequency are pre- 
sented at the ears and the intensities are kept constant and equal. 
This "effect" has been known for a number of years; a review 
of the early literature is given in the Physical Review, IX, 19 17, 
p. 502. 

The experiments recorded in the article just cited show clearly 
that the angular displacement of the apparent source of the fused 
sound or "image" is strictly proportional to the phase difference 
at the ears, with, of course, the limiting provision that the linear 
relation is true only for a difference of phase, <p y less than 1 8o°. 
(See discussion of phase angle in Section 3.7.) At <p — 180 the 
image crosses from the maximum angular displacement on one 
side of the median plane to that on the other side. The experi- 
mental procedure was to ascertain this linear relation between 0, 
the angular displacement, and <p, the difference of phase, for a 
single frequency. If, as stated, 6 is proportional to <p for any 
given frequency, then the interest centers upon the variation of 


— (which is constant at one frequency) with frequency. The 

accompanying three curves in Fig. 11.1 show the results obtained 
with three different observers. The circles and dots in the ob- 
servations for the upper curve indicate two different observational 
methods. The observations for the other two curves are shown 
by squares and triangles. 

The conclusions * from these curves are the following: 

* The binaural phase effect is being actively studied by both psychologists and 
physicists, using both sustained and impulsive sounds. It is impractical to present 
here all of the important results. While all workers may not be wholly in accord 
with the discussion in the text, the author believes these data to be reliable and the 
impression conveyed essentially correct. The data have been verified by others. 



(1) There is not a wide variation in individuals. 

(2) If these straight lines went through the point " 0," it could 
be stated (not here proved *) that the apparent position of the 
source indicated by is dependent only upon the difference in 
time of arrival of like phases at the ears. It can be claimed, 
therefore, that this conclusion as to time difference is approxi- 
mately correct. 

(3) A consideration of the computed phase differences at the 
ears with the source of sound at any given shows that the above 
quantitative measurements fully account for the ability of the 
individual to locate the source of sound in the limited region dis- 

The second of the foregoing conclusions leads to interesting 
considerations. If a source is actually placed at an angle from 
the median plane, and if it emits several frequencies, all of these 
will have the same difference in time of arrival at the ears and 
hence, according to the second conclusion above, all would appear 

* See "Acoustics," Stewart and Lindsay, p. 229. 



to the auditor to come from the same direction. This is in accord 
with experience. 

The third conclusion is very significant. It has been shown 
above that the binaural intensity effect cannot account for the 
ability to locate sounds. It is now shown that the binaural phase 
effect can do so in the limited region here discussed. There are 
limitations to this conclusion as will now appear. 

Measurements of the frequency limit of the phase effect was 
made upon 16 observers. The values of frequency above which 
no phase effect existed are only approximate and are shown in the 
accompanying Table VII. Above each frequency limit there was 
no rotation whatever of the apparent source about the head with 
changing phase. Only frequencies less than 2,000 cycles were 

Table VII 


1360 d.v. 


















1 146 




1 145 









Mean 1260 

There are two striking indications to be found in the table. 
The first is that the frequency limit is approximately the same 
for all individuals, and second, that there are exceptional wide 
variations from the mean value. The average deviation from the 
mean is 155 cycles. Omitting two observers, it is only no cycles. 
This constancy has a distinct bearing upon the conclusion that 


phase difference is the most important factor in localization up 
to 1,200 cycles. 

Subsequent experiments * have shown that the phase effect is 
not limited to frequencies less than 1,200 cycles. In fact, with 
a few selected observers of considerable experience the binaural 
phase effect has been found to exist at frequencies of several 
thousand cycles. Nevertheless the frequencies below 1,200 cy- 
cles are evidently important in localizing ability. Other factors 
may become prominent at higher frequencies. This point is dis- 
cussed in a later section of this chapter. 

Psychologists are familiar with the influence of intensity-dif- 
ference upon localization and this phenomenon is subject to gen- 
erally accepted principles. But the recognition of a phase differ- 
ence at the ears with the two intensities equal means, it would 
seem, a response to a different and more intrinsic feature of the 
stimulus. The suggestion that we have here to do with a response 
to the character of the stimulus will doubtless be regarded with 
skepticism, and in fact an attempt has been made by some to 
explain the "phase effect" in other terms. Fortunately it has 
been possible to get what seems to be conclusive evidence that 
any explanation in terms of physical intensities at the ears cannot 
be correct. This evidence is direct and readily understood. It 
has been shown above that with some individuals there are fre- 
quency regions or bands wherein the observers are not influenced 
in their localization by variations in the ratio of intensities at the 
ears, phase-difference remaining constant. With them, in this 
"lapse-region," the apparent source of the fused sound remains 
stationary in the median plane when the ratio of intensities is 
altered widely. But the significant fact is that with six t of sixteen 
observers the "phase effect" is continuous in at least a portion of 
this lapse-region. In short, the phase-phenomenon seems to be 
independent of the intensity displacement effect. The evidence 
may be found in the comparison of Tables VI and VII. For 

* Halverson, American Journal of Psychology, 38, p. 97, 1927. There are also 
unpublished results by others. 

t The reason for only six is that the lapse-regions were too high to be within 
the frequency limit of the phase effect. The phase effect was always entirely 
independent of the lapse-region. 


example, note that G. W. S. and G. R. W. have the phase effect 
in a region where the intensity effect is entirely lacking. 

1 1.3. Phase Effect with Complex Tones. — As shown above, 
the phase effect is effectively a difference in time of arrival at the 
ears, and hence the angular displacement is independent of fre- 
quency. This means that, so long as the phase difference of any 
overtone does not exceed 180 , all the tones will have the same 
angular displacement and hence there will be no confusion as to 
location. (A confusion in a horizontal semicircle only is being 
discussed.) This leads to the utilization of the phase effect. 

1 1.4. Utilization of the Binaural Phase Effect. — The binaural 
difference of phase effect was utilized during the war for the loca- 
tion of submarines and of airplanes. Obviously, if attachments 
can be made to the ears which will virtually separate them fur- 
ther, then a small rotation of the apparatus will mean a larger 
difference in phase at the receivers than at the unaided ears. 
Thus a very high accuracy may be obtained. A few observers 
seem to locate the source of sound in the rear instead of the front, 
but this does not vitiate the method. 

1 1.5. Complexity of Factors in Actual Localization. — One 

might conclude that, since the only physical factors in a pure tone 
of a given frequency are phase and intensity, and since we have 
all but eliminated intensity as a factor in localization below 1,200 
cycles, the only important factor left is that of phase difference. 
But this cannot be true in the sense that the phase difference is 
produced merely by a single source and the diffraction about the 
head as a sphere. For there are always present reflecting sur- 
faces which are extensive enough to produce images. This is 
especially true of frequencies higher than 1,000, having a wave- 
length less than 34 cm. Although the effect at the opening of 
the external meatus is still expressible in phase and intensity, yet, 
in contrast to the case of a simple source, we have, in general, the 
equivalent of several sources, with most of them on the same side 
of the median plane as the source. There would result from these 


reflections an apparently diffused source of sound instead of the 
original source only. Consequently the observer could distin- 
guish between a location on the right and left side of the median 
plane. Thus, assuming that phase difference is the most impor- 
tant factor in localization, it by no means follows that the case is 
as simple or needs to be as simple as that of a source and a rigid 
sphere with the two ears located diametrically thereon. The 
complexity of conditions involving reflection gives the single fac- 
tor, phase difference, a greater opportunity to secure accurate 
location than did the simple theoretical case. Doubtless there 
are other factors which enter into the localization of a pure tone. 
There are at least two additional ones in the case of a complex 
tone; they are the difference in quality at the two ears produced 
by diffraction, and the variation in the quality of a sound that 
depends upon the location of the source in the particular environ- 
ment; an example of the latter is one's ability, in familiar sur- 
roundings in a home, to tell from which room a voice comes. 

A suggestion has been made by Hartley and Fry * that the 
observer may have an appreciation of the distance of the source. 
Observers, however, agree that with pure tones no such appre- 
ciation exists. 

It should be noted that the discussion in this chapter of locali- 
zation of sound is limited to the apparent position of the source 
in a horizontal plane with not more than 90 . The whole prob- 
lem of localization has therefore been merely touched upon. 

1 1.6. Demonstration of Binaural Phase Effect — If two 
tuning forks of almost the same frequency, producing a "beat" 
say every five to ten seconds, are held one to each ear, the hearer 
will observe the phase effect. The phantom source of sound, 
which seems to be a single source, moves continuously about and 
in front of the head in an approximately circular horizontal path, 
but does not complete the circle. Having reached a point almost 
directly opposite the ear it moves abruptly from one side to the 
other but continuously across in front of the observer. 

* Physical Review, 13, 191 9. 


xx. 7. Binaural Beats. — There is an interesting phenomenon 
that may easily be observed. If two beating tuning forks are 
held one to each ear, the beats can be heard. In listening to two 
beating tones with one ear the combined intensity varies from a 
maximum to zero. With binaural beats the minimum intensity 
is distinctly not zero. Moreover, if one listens closely, he can 
hear two additional swells of intensity, one just before and one 
just after the minimum intensity. These additional or secondary 
maxima are present only if the beat period exceeds at least one 
second. The phenomena involved in binaural beats will, when 
fully investigated,* doubtless increase the understanding of au- 


1. Why cannot the phase effect be explained as an intensity effect? 

2. Why cannot phase fully account for localization ? Why cannot 
the phase and the intensity effect together fully account for it? 

3. From the data in Fig. ii.i, compute the time difference that 
can be detected, assuming that one may notice the variation of from 


* For a report on binaural beats with a record of new phenomena see Stewart, 
Physical Review y IX, No. 6, June 1917, p. 502, 509 and 51 4. The most interesting 
conclusion in these papers is, that the evidence points to the existence of a second 
and a different organ of hearing, the saccule. Recent experiments verify this 
conclusion. See also a later discussion by Lane, Physical Review y 26, p. 401, 
1925, and by Stewart, Journal of Acoustical Society of America^ April, 1930. 


i a. i. Transmission of Energy from One Medium to Another. 

— In previous chapters there have been discussed several cases of 
transmission where the medium remained the same and yet, be- 
cause of the changes in the confinement of that medium, the 
energy transmitted was not ioo per cent. In Section 5.7 it was 
found that there was a reflection in a conduit at any sudden 
change in area, indeed, also when a conduit opened out into the 
unconfined atmosphere. In Section 6.9 the reflection at the open 
end of megaphones was mentioned. But the transmission of 
sound from a gas to a solid and from one solid to another involves 
new considerations. The physical factors entering the question 
of such transmission from one medium to another will now be 

In the first chapter it was made evident that the transmission 
of a sound wave depends upon the elasticity of the medium and 
also upon the density. The former quality requires that the dis- 
placement return to zero value; it gives a return force which is 
essential in the process. The density indicates the existence of 
mass and hence of the requirement of time to produce a displace- 
ment. Both elasticity and density are requisite. Without either 
the wave would not be produced. It is therefore to be observed 
that in the value of the velocity of a sound wave in a fluid there 
are involved several physical factors. It may then not be sur- 
prising that in a plane wave, incident perpendicularly at a plane 
interface between two media, the percentage of energy in the 
transmitted wave depends upon the product of the density and 
the wave velocity in each medium. This product is called the 
"acoustic resistance," the second word being used also in elec- 
tricity, but not in a closely analogous manner. Consider the 
plane interface, one medium being on the left and one on the 


i 5 o 


right. Let the incident wave come from the left. Of the energy 
flowing to the right, part will be reflected at the interface and the 
remainder will be transmitted into the second medium. It is for- 
tunate that the actual amount transmitted can be determined by 
the application of the following simple expression: * 

Transmitted flow of energy 
Incident flow of energy 


(r + i) 2 

Here r is an abbreviation for the ratio of the acoustic resistance 
in the second medium to that in the first. 

It is also an important fact that this discussion can be ex- 
tended to include solids if one limits the consideration to longi- 
tudinal waves. This would include the passage of sound from 
air to water and from water through a ship's hull.f It would be 
applicable to the transmission of sound from air to the ground 
and vice versa. But one must be warned that it is only applicable 
where the second medium does not act like a drum head or a dia- 
phragm. In fact, in partitions and floors, diaphragm action is 
nearly the correct description, for it is found that mass is a very 
important factor. The actual sound entering a wall, treated as 
a medium as in this chapter, would be small indeed and this does 
not agree with the amount of transmission found in experience. 

The following table gives the value of the acoustic resistance 
for several substances: % 


40 X io 6 

Cast iron 



*9 :: 






.029 -.066 ' 



* For the derivation see Stewart and Lindsay, "Acoustics" (D. Van Nostrand), 
Chapter IV. 

t Not precisely true, for the diaphragm action discussed in this section may not 
be disregarded. 

X See Appendix I of Stewart and Lindsay, "Acoustics," p. 327. 


As an example, if one applies the formula to the transmission 
from air to water, a very small fraction is found. 

12.2. Transmission in Architectural Acoustics.* — From the 
foregoing one might correctly conclude that the structure of build- 
ings to avoid transmission of sound from one room to another is 
not a very simple matter. Interesting experiments have been 
made with partitions in order to find the most economical con- 
struction that will give satisfactory acoustic insulation. The 
lighter the structure, such as in a home, the more difficult becomes 
the problem. For in a building requiring massive walls and floors, 
the inertia prevents diaphragm action. It is to be borne in mind 
that to reduce the sound transmission through a floor or partition 
there are chiefly two methods, absorption and rigid or massive 
construction. The absorption can be produced by the nature of 
the material itself or of its surface. This will prevent, through 
surface absorption, the transmission of sound, but the absorption 
on the interior is not so effective if the wall is light enough to 
vibrate as a diaphragm. Both mass and rigidity of a wall or floor 
will prevent transmission. An illustration of the use of absorp- 
tion and rigidity is as follows. An ordinary wood floor in a home 
consists of the joists carrying two layers of flooring, the top one 
being the finished floor. Underneath the joists are the lath and 
the plaster. Not only will such a floor vibrate as a whole, but 
even the areas of flooring from one joint to another will have 
additional vibration. One way of producing a serious reduction 
of the transmission is to lay over the first flooring an absorbing 
material of perhaps an inch in thickness. Upon this, and sepa- 
rated by perhaps six inches, are laid 2 X i's but without nailing. 
Then the finish floor is nailed thereto. Thus the finish flooring 
floats without any solid connection with the floor structure. The 
effect is very marked. 

12.3. Machinery Noises. — Noises from machinery may be 

prevented by the removal of the cause, by absorption and by 

* In this country there is an organization of physicists, acoustic engineers, 
architects, and construction engineers, the Acoustical Society of America, that is 
actively interested in all problems relating to architectural acoustics. 


preventing the flow of energy from the machine to surrounding 
supports. Attention here will be devoted especially to the last 
method for the others are more obviously applicable. Assume 
that the desire is to prevent the 120 cycle hum of a motor from 
being conveyed to a support such as a table or floor. Everyone 
knows that a soft pad placed under the motor will be quite effect- 
ive in preventing the transmission. This is not merely because 
of the absorption of the pad but also because of its elasticity. 
One can substitute a number of small springs and get a good effect 
also. This is because the springs cause a reflection of the energy, 
explained briefly as follows. The presence of the springs will 
allow the motor base to vibrate rather freely. There will then 
exist a condition much like that at the open end of a pipe, previ- 
ously discussed. In other words the condition is one of a reflec- 
tion. This will be particularly true if the springs are light and 
numerous, rather than heavy and few. The mathematical treat- 
ment gives a much better explanation,* but the chief point to be 
realized is that we are here dealing with a case of reflection of 
energy, rather than of absorption. Attention of engineers to the 
prevention of machinery noises is rapidly increasing.f 

12.4. Case of Three Media. — Assume that we are dealing 
not with the transmission from medium one to medium two, but 
also from medium two to medium three, retaining perpendicular 
incidence of the sound and parallelism of the two planes separating 
the media. Then, if this second medium has a length such that 
resonance is obtained through the reflection at the two interfaces, 
there is a curious result. The transmission of energy from the 
first to the third medium is then of the same magnitude as would 
exist if the second medium were absent. Usually in order to 
ascertain the energy transmitted in the third medium it is neces- 
sary to use a complicated formula. 

* Kimball, Journal of the Acoustical Society, II, 2, 297, 1930. 
t See Slocum, " Noise and Vibration Engineering," D. Van Nostrand Company, 



12.5. Constrictions and Expansion in Conduits. — (1) Inten- 
sity Effect. — It has been explained in a previous chapter that if 
the area in a tubular conduit is changed at any point, there is not 
100% transmission, but a reflection. If there are two changes in 
area as illustrated in the accompanying Fig. 12.1, then there are 






Fig. 1 2. 1 

reflections at both junctions where the areas are altered. It is 

assumed that the diameters of conduits 1, 2 and 3 are small in 

comparison with a wave-length. If the area of the first tube is 

S 2 
Si f of the second 4S2, and of the third £3, and if we put mi = y 

and m 2 = -jr > then the following conclusions may be drawn: 

(a) If the length of the tube area S 2 is very short compared 
.with the wave-length, then the ratio of transmitted to incident 
energy depends only upon the first and third tubes and may be 
expressed by 

^m 1 m 2 4$iSz 

(mim 2 + i) : 


(Si + S 3 ) ! 

(i>) If the tube S 2 has a length equal to one-half wave-length, 
then there is resonance set up in S 2 and the transmitted wave is 
the same as if & were not present. This is also true if *SVs length 
is any integral number of half-wave-lengths. 

(c) The ratio of transmitted to incident energy is known for 
any length of oY 

The reader is now enabled to understand why it is difficult to 
diminish transmission in a conduit by a constriction which is very 
short in length. An example would be the insertion of a dia- 


phragm across the tube with a small hole in it. According to 
item (a) just stated, since £1 and £3 would be equal, the entire 
incident energy would be transmitted. Of course viscosity and 
the assumption of plane waves in the theory would prevent the 
accuracy of this statement, but its truth is sufficiently approxi- 
mate as a practical guide. If one attempts to pinch a rubber 
tube conduit and thus to reduce the intensity transmitted, prac- 
tically no change in intensity will occur until the channel has been 
made very small. The ear canals may be nearly closed with im- 
pacted wax but no deafness will be noted until there is practically 
complete closure. 

(2) Phase Effect. — At first thought it might seem that, al- 
though a constriction or expansion will change the intensity of 
flow of energy in a conduit, yet there would be no change of phase 
other than that which would ordinarily occur in the same length 
of tubing of constant diameter. But this is not the case, for there 
are repeated reflections at the ends of the constricted (or ex- 
panded) length and the transmitted wave is made up of not merely 
a portion of one wave incident at the S 3 end of the constriction, 
but of a large number. Its phase cannot be unmodified by this 
complexity. The theory for this change is known.* 

12.6. The Stethoscope. — Consider an ideal stethoscope as 
shown in the accompanying figure. Medium 1 is a solid, liquid 
or any medium. The medium in 2 and 3 is air. The query 
arises as to the transmission of energy from medium 1 into the 
small tube. The theory is known f and the following conclusions 
may be drawn from it: 

I. If the thickness of the lamina 2 is very small in comparison 
with a wave-length, the ratio of the transmitted energy to the 
energy incident in 1 is 


(n + mi) 


* See Stewart and Lindsay, Acoustics, p. 78. 
t Briilie, Le Genie Civil, 75, 223, 1919. 



Fig. 12.2 

where r\ is the ratio of the acoustic resistances of medium 2 to 
medium 1 and m* is the ratio of the area of 3 to the area of 2. 

2. If one considers the transmission from water to air, and 
considers the thickness of medium 2, he will find that this length 
can readily be adjusted 
so that the energy of 
sound in tube 3 is much 
greater than the energy 
which would pass from 
the given area of water 
to air. Moreover, the 
stethoscope tube permits 
all of the energy in tube 
3 to flow into the ear, 
whereas without the 

stethoscope only a small part of the flow of energy from the given 
area of water would enter the ear. When the ear is pressed 
against medium 1, it is obvious that the ear becomes a stethoscope. 

12.7. Non-reflecting Conduit Junctions. — It is impossible to 
change the area of a conduit without introducing reflection. In 
quantitative measurements this is a serious consideration. In 
order to avoid reflection it is customary to connect the two differ- 
ent areas by means of a cone of very gradual slope.* The longer 
the cone, i.e., the less its slope, the less the reflections at the ends. 
Also, since the reflections at the ends of the cone introduce the 
possibility of resonance, all wave-lengths will not be transmitted 
equally well. Only at resonance will the ratio of transmitted to 
incident energy be unity. In many cases, the introduction of felt 
in the transmitting tubes will cut down resonance and prevent 
any material effect of reflected waves upon the source and thus 
will make possible satisfactory quantitative measurements of the 
relative intensities of different frequencies. 

12.8. Velocity of Sound in Pipes. — There are two physical 
phenomena which enter into sound transmission in pipes but 

* See also Section 5.7. 

i 5 6 


which are not important in the open air. The viscosity of a gas 
is made prominent in a pipe because of the presence of the sta- 
tionary wall which necessitates slippage in the gas itself. More- 
over, a gas is heated by compression and cooled by rarefaction. 
The exchange of heat between the wall and the gas is the second 
factor. The result of both of these factors is to diminish the 
sound intensity and also the sound velocity. It is found experi- 
mentally that the percentage decrease in the velocity of sound in 
a pipe is given by the following expression. 


wherein r is the radius of the pipe in mms., n is the frequency 
and c a number which varies with the diameter and the material 
of the pipe. Schulze * in 1904 found experimentally that c varied 
from .0075 to 0.025. 

12.9. Decay of Intensity in Pipes. — The decay of intensity 
of sound in transmission through pipes has been only slightly 
investigated. It is well-known that the percentage of decay per 
unit length should be the same throughout the pipe and should 
be dependent upon the frequency, the diameter and the material 
of the pipe. H. Brillie t presents the following data as having 





5 mm. 
15 mm. 
10 mm. 

165 cm. 

330 cm. 

1575 cm. 

been taken by Messrs. Clerget and Dessus in France. The 
lengths of pipe given are those required to reduce the sound in- 
tensity to 50% of its initial value. The frequency of the sound 
used in these experiments is not given by Brillie. 

The accompanying Fig. 12.3 graphically exhibits the data ob- 

* Schulze, Ann. d. Physik, 13, p. 1060 (1904). 
t Brillie, Le Genie Civil, 75, p. 224, 1919. 



tained with transmission through speaking tubes by Eckhardt, 
Chrisler and Evans.* 

The loss in transmission depends importantly upon the diam- 
eter of the tube. Of course the transmission in a 20-foot tube 





1200 1800 2400 

1 U IR0N 

L I *=i ~l"FIBRE 


Fio. 12.3 

would be the square of the values shown. Thus a 2-inch brass 
tube would transmit 80% of a tone of 300 cycles in a 10-foot 
tube, 64% in a 20-foot tube, and 51% in a 30-foot tube. Thus 
in long tubes the differences in Fig. 12.3 are accentuated. The 








E m 








600 1200 1800 2400 3000 

Fio. 12.4 

loss in transmission also depends upon the material of the pipe. 

But it is not clear whether the difference is caused by friction or 

by absorption due to the lack of perfect rigidity in the pipe. 

The passage of sound around a bend in a pipe has already been 

* Technological Paper of the Bureau of Standards No. 333, p. 163, Vol. 21, 


mentioned in Section 10.18. The authors just mentioned have 
determined also the effect of 90 bends. The results are shown 
in Fig. 12.4. There is an unexplained peculiarity with high fre- 
quencies with a pipe of large diameter. 


1. According to (2) of Section 12.6, how does the excess pressure 
in the tube containing medium 3 compare with the pressure which 
would be obtained if the tube containing 2 were extended indefinitely 
to the right and the former tube omitted? 

2. Explain the apparent contradiction involved in the following: 
According to Section 12.1, the transmission of sound energy from 
medium 1 to medium 2 of Fig. 12.2 depends upon the ratio of the 
acoustic resistance in the second medium to that in the first. Then 
the attachment of the tube containing 3 could not increase the flow 
of energy from 1 to 2, and hence the energy computed as in 1 2.6 can- 
not be correct. 

3. Compare Sections 12.7 and 5.7. For a wave-length long com- 
pared to the abrupt changes proposed in 5.7, there would be little 
difference in result between the long cone and the approximation to 
it by a succession of short cylindrical tubes connected by abrupt 
changes in area. From this fact what would you conclude concerning 
the reflection on the walls of the cone as well as at the ends? 

4. Assume a conduit had walls that were thin and elastic so that 
they would be set in vibration by the sound w % ave. Can you venture 
a reason why such walls would modify the velocity of the sound wave 
in the tube? 


13.1. Interference Tube of Herschel and Quincke.*— In 

transmitting sound through a tube or conduit, it is often desirable 
to eliminate certain frequency regions and this chapter describes 
the methods so far devised to produce such a selection. Consider 
the transmission of sound, from left to right, through the double 
tube shown in the accompanying Fig. 13. 1. 

Fig. 13.1 

As already noted in Chapter V, if the area of cross-section of 
tube AD is twice that of either DBE or DCE, then a wave passing 
from A to D will suffer no reflection at D but will divide equally 
and pass on to E. If DBE and DCE are alike in length and area, 
and if EF has the same diameter as AD, the combined wave at E 
will pass on through EF without reflection at E because there is 
no change in condition. But if DBE is, for a given frequency, 
one-half wave-length longer than DCE, then the two waves will 
meet at E out of phase. When the pressure of one is positive 
the other will be equal and negative, but the displacements will 
unite favorably for a positive displacement of one is the same 
actual direction as the negative displacement of the other. The 
pressures neutralize, but the displacements add. There can be 
no forward wave in EF for there is no pressure to produce it. 

* See Rayleigh's "Theory of Sound," Vol. II, pp. 64, 65 and 210. 



The energy cannot be destroyed by interference; the two waves 
proceed in their journeys around the loop through C and B and 
unite at D. Since the difference in paths is now zero, the pres- 
sures are in phase and the waves proceed from D to A. Thus, 
assuming the conditions specified, there will be no transmission 
of the energy through EF. If the sound entering A is not com- 
posed of a single frequency but is complex, that frequency for 
which the difference in path is one-half wave-length will be elimi- 
nated from transmission. 

But the above explanation, although apparently satisfactory 
to physicists for almost a century (from 1833 to 1928 *), is in error 
in the inference that elimination will occur only for frequencies 
which are opposite in phase after passing over the paths DBE 
and DCE. The complete theory of the Herschel and Quincke 
tube shows that there are other frequencies, in fact, from two to 
three times as many, which will fail in transmission through EF. 
The secret of their appearance rests in the fact that, in general, 
the history of a sound wave leaving D is not simply passage from 
D to E by the two paths. One can see that the wave travelling 
by the path DCE will divide at E and part go out through EF, 
part through B back to D where it will again divide and part will 
be reflected back through C to D, etc. At these division points 
there will be in general a change of phase with the reflected waves. 
It is obvious that one cannot actually trace the paths of the waves 
in intensity and phase in the general case for they will not be 
limited by one circuit about the loop. Thus it becomes necessary 
to resort to mathematical methods of stating the conditions which 
must exist at D and E. When this is done and the equations 
solved, the additional eliminated frequencies are discovered. 
They are found to depend upon the sum of the lengths of the two 
branches rather than upon their difference. There is zero trans- 
mission when the sum of these lengths is an integral number of 
wave-lengths, provided that at the same time the difference of 
these lengths is not an integral number of wave-lengths. This is 

•Stewart, Phys. Rev., 31, 4, 696, 1928, or Stewart and Lindsay, "Acoustics," 
p. 90. 


an interesting case of the advantage of mathematical methods. 

Quincke has made use of a modification which is more simple 
in construction. The wave enters at A, Fig. 13.2, and either 
passes out at F or experiences what is equivalent to a reflection 
at D, for there is no opportunity for the energy to be disposed of 
otherwise. If the frequency corresponds to the natural period of 
vibration of the tube DC, then this tube will resonate. As noted 
in a previous chapter at reso- D 

nance frequency the incoming and a > Z 

outgoing waves in such a closed 
tube agree in displacement at the 
open end, but have pressures 
that are approximately equal and 
opposite, forming a "loop" at ^ C 

the end. If this occurs, there is 

approximately no pressure to produce transmission out through 
DF. Thus the wave of this frequency is eliminated from trans- 
mission. Since the elimination is caused by resonance, the shape 
of the tube DC is inconsequential, if the elimination of this one 
frequency only is considered. 

But a more detailed description of the action in Fig. 13.2 will 
make the phenomenon clearer. Consider the number of possible 
waves involved. There is the flow of energy from A to D, which 
is wave one. There are the two waves from wave one, one enter- 
ing DC described as two, and one passing on toward F, described 
as wave three. (For simplicity let us regard the reflection in pass- 
ing into and out of DC as nil. We are not so much interested in 
an accuracy of treatment as in a further helpful description.) 
There is one wave from the tube DC which will pass partly to the 
left toward A and partly to the right toward F. Denote this 
wave in the tube by six, the one passing toward A by four, and 
the one toward F by five. We then see that there are two waves 
to the right, toward F, numbered three and five. As hereinbefore 
noted, time is required to build up resonance, and, if there is no 
viscosity, there is only the limiting case of vibration in DC where 
the energy escaping from the resonator is equal to that entering 

1 62 


from the source. The details are as follows. Resonance in DC 
will build up until five is as great as three. But five is opposite 
in phase to three because it has traversed a half wave-length fur- 
ther. Wave five will not build up any further because at equality 
of three and five the total flow to the right becomes nil. Then 
the total flow in four must become equal to the flow in one. In 
other words, wave one is essentially reflected. This explanation 
gives a better insight into the phenomenon, though we have in- 
correctly neglected the reflections occurring in passing in and out 

Fig. 13.3 

of DC and the alteration of phases involved at this point. Even 
with these reflections we will never have any greater number of 
waves than those specified, though they arise from more causes 
than described. With resonance without viscosity one sees that 
wave three may equal wave five and wave four may equal wave 

As illustrations of possible shapes, two arrangements are shown 
in Fig. 13.3. But, as shown below, the shape of this tube does 
determine the extent of the partial elimination of neighboring 

An investigation of the theory (see the following section) 
states that if it were not for viscosity, the elimination of the 



selected frequency would be complete and independent of the di- 
ameter of this side branch. In practice one branch will eliminate 
all but a fraction of the incident energy. Consequently more 
than one branch is sometimes essential to the production of the 
desired reduction in intensity. Krueger * has studied the use of 
such branch tubes and has concluded that the extent of the elimi- 
nation depends upon the point of attachment of the side tube. 
He used a fork resonator and found that the elimination was the 
greatest when the point of attachment to the conduit was an inte- 
gral number of half wave-lengths from the rear wall of the reso- 
nator and the least when the distance was an odd number of 
quarter wave-lengths. Undoubtedly these conclusions indicate 


33.2 — «-H- 33.2 

**- 33.2 


Fig. 13.4 

the influence of the reflected wave on the source, but a discussion 
of the matter should be based upon additional experiments. It 
is not difficult, however, to understand that there is a desirable 
separation of side tubes when using more than one. As an illus- 
tration, attention is directed to Krueger's final design, shown in 
Fig. 13.4. It was constructed to eliminate a frequency of 256 
cycles or an integral number of times this frequency. 

Side tubes A, C and F will assist in eliminating 256 cycles per 
second if each is adjusted to have a length of one quarter of the 
wave length; A, B, C, D and F y 512 cycles if correspondingly 
adjusted; and similarly A, B, C, D, E and F, 1,024 cycles. The 
object of the spacing is to cause the reflected waves to be in agree- 
ment. Consider the fact that the incident and reflected waves 

* Krueger, Philos. Stud., 17, 1901, p. 223. 



at each opening are opposite in the phase of the displacement. 
Assume a wave incident at the right in Fig. 13.4. If the distance 
B to A is one-half wave-length, then the phases of the waves re- 
flected at the two points are opposite (for the incident waves are 
opposite in phase). But, since the reflected wave must travel 
from B to A, a half-wave-length, it will there be in the same phase 
with the reflected wave at A. Consequently there will be no 
interference in the reflected waves, a fundamental condition for 
the maximum elimination. 

13.2. Theory of a Closed Tube as a Side Branch. — A the- 
oretical investigation by Stewart * considers a single side tube, 
and only the incident and transmitted waves. The wave reflected 

at the opening of the side 
tube is assumed not to 
affect the source. This 
condition can be approx- 
imated by having a 
considerable amount of 
"damping," e.g., hair- 
felt, distributed along 
the conduit between the 
source and the branch 
tube. Also it is assumed 
that the transmitted 
wave is not reflected at 
the distant terminus of 
the tube. This ideal con- 
dition can be approxi- 
mated by again inserting 
damping in the conduit 
between the side tube 
and the distant terminus. 


700 8009001000 

Fig. 13.5 


Obviously the damping will greatly dissipate the original energy, 
but the arrangement will permit a comparison of the theory of 
* "Acoustics," Stewart and Lindsay, p. 126. 


the action and the experimental results. The comparison is 
found in Fig. 13.5. The full line curve gives the theoretical 
values * of the square root of the transmission. 

13.3. Hehnholtz Resonator as a Side Branch. — Here the the- 
ory and experiment bring rather unexpected results for the re- 
sponse of such a resonator in the open is "sharp," that is, fre- 
quencies other than the critical one produce very little vibration. 

2 3 4 5 6 7 8 9/0 





Fig. 13.6 

* The theory shows that the ratio of the transmitted energy to the incident 
energy is 

r , a* tan 2 M 2 "]" 1 


where a is the area of the branch tube, S the area of the conduit, k is 27r divided by 
the wave-length and / is the length of the side branch. At the critical frequency 
tan kl = °° , and the transmission is zero and independent of the areas of the tubes. 
The function "tan kl" is a short expression used in trigonometry for indicating a 

certain value that depends upon the angle kl or , where X is the wave-length. 


The formula is given here merely that the reader may see that a result complicated 
physically may have a simple mathematical expression. 


The theoretical * and experimental results are shown in Fig. 13.6, 
the former by the graph and the latter by the circles. 

It is to be observed that the transmission is affected very 
markedly over a wide range of frequencies. The Helmholtz reso- 
nator would not be efficient as a Quincke tube if only one fre- 
quency or a narrow range is to be eliminated. 

13.4. Action of an Orifice. — In this connection the action of 
an orifice in a conduit is interesting. It might be supposed that 
the sound escapes from the orifice and thus diminishes transmis- 
sion. But this is not the correct picture. Figure 13.7 shows a series 
of curves taken with orifices of four different diameters. The full 
line curve is the theory f and the designated points represent the 
square root of the measured values of the transmission. The 
ordinates on the right give the values in decibels. Curve 4 does 
not agree with experimental points. Curve 5 is the theoretical 
curve if cq be arbitrarily changed from its computed value 0.582 
to 0.74. 

A point of interest is that the orifice affects the low frequen- 
cies the most. However, this conclusion cannot be extended to 
indefinitely high frequencies. Two points which can be seen only 
through theoretical considerations are that the cause of the de- 
creased transmission is more importantly the reaction of the mass 
in the orifice and that the viscosity of the orifice is relatively 
unimportant. The former point needs explanation. Of course 
there is a radiation from the hole outward and this energy is lost. 
But because of the action of the inertia of the mass of the gas in 
the hole, the reflected wave, similar to that discussed earlier in 
this chapter, is very much greater than the radiated wave and 
consequently is the chief factor in the resulting decrease of trans- 

* The theory, Stewart and Lindsay, "Acoustics," p. 116, shows that the ratio 
of transmitted to incident energy is |i + 4^ I A j . Here the sym- 
bols not defined in the foregoing are c the "conductivity" of the neck of the 
resonator and V its volume. 

fThe theory, Stewart and Lindsay, "Acoustics," p. 120, is too complicated 
for a brief footnote. 



mission. However the relative importance of the radiated wave 
increases with frequency. The above interesting points can be 
applied to a musical instrument like a flute, but not without addi- 
tional discussion, for here we have standing waves. 

It has just been stated that the reaction of the mass is very 
important, indeed more so than the radiation from the orifice. 

Again the mathematical theory is the only adequate description, 
although one may perhaps wisely attempt a further discussion in 
language. In the early chapters the transmission of sound was 
seen to be possible because a medium possesses inertia (or mass) 
and elasticity (or the ability to return to the original position). 
In a tube or conduit having a constant cross-section there is no 
reflection backward. The reason is that these two qualities, 
inertia and elasticity, are the same throughout. If one can imag- 
ine the air losing its elasticity at a certain point, then there would 


be no pressure there and there would be reflection at that place 
approximately as at the open end of a pipe. In short, were the 
medium at any point along the tube to lose its elasticity and have 
inertia only, there would be reflection. Now an orifice acts very 
much like a medium having inertia only. Indeed, in the Helm- 
holtz resonator, the orifice has already been assumed to have 
inertia only. This was because it led into a large volume, the 
elastic action of which was much more important. The experi- 
ment with the Helmholtz resonator illustrates that the orifice, 
although effectively so short, but opening out into space, may act 
as if it possesses inertia only. Thus the reflection may prove to 
be important and the transmission through it is small in compari- 
son. Experimentally (and theoretically also) the radiation is not 
the important element. In fact, when one raises a key on the 
flute or clarinet it is not primarily for the purpose of allowing 
energy to escape, but rather to cause reflection with change of 
pressure phase. More concerning the action of such instruments 
will be given in Chapter XV. What is here stated is but one 
aspect in a very complex acoustic condition in such instruments. 

13.5. Acoustic Wave Filters. — A very striking and effective 
means of eliminating specific ranges of frequencies has been 
found * in the acoustic wave filter. Its detailed description is 
beyond the scope of this presentation, but certain points of prac- 
tical interest will be considered. The construction of three types 
of filters is shown in Fig. 13.8. 

Type A consists of a series of Helmholtz resonators distributed 
at equal distances along a conduit, Type B of orifices similarly 
distributed, the side walls being extended to increase the inertia 
of the orifices, and Type C of the combination of the two pre- 
ceding types. The characteristic curves of transmission are 
shown in Fig. 13.9, the letters corresponding to the types. 

Type A is a low-frequency pass, Type B a high-frequency pass, 

* See Stewart, Physical Review, 20, 1922, p. 528; 22, 1923, p. 502; 23, 1924, 
p. 520; and Jl. of Opt. Soc., 9, 1924, p. 583; and Hall, Phys. Rev., 23, 1924, p. 116; 
and Peacock, Phys. Rev., 23, 1924, p. 525. Or see "Acoustics," Stewart and Lind- 
say, Chapter VII. 



and Type C a single-band pass. The remarkable property of 
these filters is an almost total elimination of transmission in the 
attenuated frequency regions. 









Fig. 13.8 

There can be a vibration of a medium to and fro without any 
transmission of power. For example, consider a standing wave 
in a tube of infinite extension. Place walls across the tube at 

Fig. 13.9 

two displacement nodes, not necessarily adjacent. The vibration 
would continue in this closed space indefinitely, were it not for 
the dissipation of energy caused by viscosity and absorption of 


the walls. A series of such closed spaces could be placed along 
the tube. One would then witness an oscillatory motion or vibra- 
tion in each* compartment, but no transmission of energy. This 
is not analogous to the wave filter but is introduced to show the 
possibility of no transmission of energy. Now theory shows that 
in the acoustic wave filter there is a region of frequencies where 
there is a vibratory motion in each "section" or space between 
dashed lines in Type A of Fig. 13.8, without a flow of energy from 
one to the other and with an amplitude of vibration constantly 
decreasing from section to section in the direction of the attempted 
transmission. This is the phenomenon in the non-pass region of 

If one wishes to set a pendulum in vibration by communi- 
cating energy throughout its vibration, the effort must be prop- 
erly timed, that is, the applied force must have a certain phase 
relation to the movement. In a similar manner in the acoustic 
wave filter, it is possible to have a phase relation between pressure 
and particle velocity such that energy will be transmitted. Thus 
theory shows the possibility of a non-attenuated transmission 
region of frequencies in the acoustic wave filter. 

From these two analogies the possibility of an acoustic wave 
filter having attenuated regions and non-attenuated region is sug- 
gested. Although this discussion does not consider the actual 
theory of the acoustic wave filter, enough has been said to indicate 
that the acoustic wave filter does not dissipate energy, but reflects 
it, thus refusing transmission. 

13.6. Fictitious " Nodes." — The discussion in the preceding 
paragraphs makes this a convenient point to mention the absence 
of ideal displacement nodes in the stationary waves occurring in 
practice. No energy could be transmitted through such an ideal 
node. For, although here is a "loop" of pressure and ample 
variation from the mean pressure to do work, yet there is no mo- 
tion of the medium and without movement a force cannot do 
work. In the case of standing waves in an open organ pipe, there 
is a flow of energy out of the open end. Thus there must be a 


flow from the mouth to this end. With a closed pipe, all the 
energy issues from the mouth, but even in this case the displace- 
ment nodes in the air column are not strictly stationary, for there 
is a loss of energy in the wave travelling along the tube. This 
loss occurs because the air next to the pipe remains stationary and 
a friction loss is introduced. Then, too, wave energy is commu- 
nicated to the walls of the pipe. The wave travelling from the 
mouth to the end of the pipe thereby diminishes slightly in am- 
plitude with distance of travel. The same sort of diminution 
occurs as the wave returns. Of course similar losses occur in the 
open pipe as well. From what has been stated, none of these 
cases can have ideal displacement nodes. This explanation of the 
existence of imperfect " nodes" must be modified by the statement 
that the viscosity loss must enter through the actual particle 
velocity of the resultant wave, rather than in the particle velocity 
which each wave, forward and back, would have if existing alone. 


1. Show that for waves returning from E to D of Fig. 13.1, the 
displacements are favorable to transmission in DA, but unfavorable 
to transmission again in the branches, DB and DC, provided the dif- 
ference in the lengths of the branches is one-half wave-length. 

2. How would the introduction of viscosity affect the considera- 
tion of the divided tubes shown in Fig. 13. 1 ? 

3. What is the reason that the Quincke tube does not give zero 
transmission at the critical frequency? 

4. Why does the presence of a mere mass of gas in the orifice pre- 
vent a serious amount of dissipation of energy out through the orifice? 

5. In both Figs. 13.5 and 13.6 the effects of the resonators are 
not as sharp as would be expected from their response in the open. 
Can you suggest a possible reason ? 

6. Why might one expect that a "large'' orifice would give a result 
not in conformity with theory as in Fig. 13.7? 


14.1. The Diatonic Scale.— There are a number of different 
musical scales employed in the world, but western countries use 
chiefly the one to be described. 

Several reasons exist for expressing the relation between the 
frequencies of two tones not as a frequency difference but as a 
frequency ratio. It is a fact that to us the most pleasing combi- 
nation of two tones is one which the frequency ratio is expressible 
by two integers neither of which is large. Thus what is called 
the octave has a frequency ratio of 2 : 1 . The manner in which 
the range of frequencies between a note and its octave is divided 
determines the nature of the scale. Our scale consists of eight 
notes, the ratios of the frequencies to the first or "tonic" being 
as follows: 















3/ 2 





5/2 etc. 

It will be seen at once that the ratios of 

C, E, and G 
F y A, and c ' 
G, B, and d' 

are all 4 : 5 : 6, which is a major chord. The notes of £, G, B 
have the ratios 10 : 12 : 15, which is called a minor chord. 

If the ratios of the frequencies of adjacent notes be now written 
we have 



9/8 10/9 16/15 9/ 8 IO /9 9/ 8 16/15 


An "interval" between two tones is the ratio of their frequen- 
cies. If the frequencies of three tones are represented by a y b y 
and c, then the three intervals are |, - and -, the last being 
clearly the algebraic product and not the sum of the first two. 
Thus the interval C to D is 9/8 or 1.125, from D to E y 10/9 or 
1. in, and the interval C to E y 1.125 X x.ui- 

There are indicated above three distinct ratios or "intervals" 
9/8, 10/9, and 16/15, or 1.125, i.m, and 1.067. The first two 
are practically equal and approximately twice the third. If the 
first is called a whole tone, the latter is a semitone. If now addi- 
tional semitones be inserted between C and D y D and E, F and 
G y G and A y and A and B y there are, as a result, 12 intervals in 
the octave. But with the above ratios these twelve intervals 
could not be all equal. Hence one could not pick out the tones 
that would produce the same scale if he began on D as a keynote. 
As an example, interval D to E is 10/9 whereas C to D is 9/8* 
Similar obstacles would arise throughout in attempting to have 
the same scale on D as a "tonic." One recourse would be to 
introduce a sufficient number of tones in the scale to make it 
possible to start on any note as the tonic. But this is not prac- 
tical. Consider the impossibility of applying it to the piano forte. 
It is evident that a sacrifice of harmony must be made for con- 
venience in execution. The modification in the tones necessary 
is called temperament. At least four suggestions as to tempera- 
ment have been made, but only one has been retained. It is 
called the mean temperament. 

14.2. Mean Temperament. — The mean temperament assumes 
twelve fixed intervals. If the scale is to be entirely independent 
of the tonic, then the twelve intervals must be exactly alike. In 
order for this to be true, each one of these intervals must be 1.059 
instead of the 1.067 S iven above. The reason for this is that 
1 X 1.059 X 1.059 (repeated, occurring twelve times) equals 
2.0.* The intervals between C and the other notes of the scale 

* This is the significance of what is meant by the twelfth root of 2.0* A semi- 
tone interval multiplied 12 times (which means 12 such intervals) equals 2.0 or the 


are shown for the "natural" and the "tempered" scales as follows: 









Natural Scale ...... 

Tempered Scale .... 

I. coo 
I. coo 

1. 125 
1. 122 







On the tempered scale C# would be 1.059 anc * would be the same 
as Db. By placing the semitones, 1.059, in between C and Z>, 
D and E, F and G, G and A> A and B, we have twelve semitones 
in the octave and any one of them may be used as the tonic. In 
the actual tuning of a piano care is not taken to secure this equal 
temperament, but it is approximated. These introduced semi- 
tones would have the following intervals with the tonic: 

C# and Db 1.000 X 1.059 

Z># and E\> 1.122 X 1.059 

F# and Gb 1.325 X 1.059 

G# and A\> i. 49 8 X 1.059 

AH and B\> 1.682 X 1.059 

The difference between the natural and the tempered scale is 
slight and scarcely noticeable if the tones are played in succession. 
But in chords, it is said that the difference is very noticeable (see 
Section 14.6). 

14.3. Frequency. — There is not universal agreement as to the 
frequency of a given note. The following can be found men- 
tioned in various works in acoustics: 

Standard Frequencies for A 

French 435 

Stuttgart 440 

Concert Pitch 460 

International 435 

American Concert 461.6 

Boston Symphony Orchestra 435 

According to White * all piano manufacturers of the United 
* White, Science, 72, No. 1864, p. 295 (1930). 



States are using an A of 440 vibrations. With A 440, middle C 
becomes 261.6 on an equally tempered scale. 

14.4. Nomenclature. — There are various ways of indicating 
the octave in which a given note is found. Two examples follow: 
C-i to B-i; Co to B ; G to B\\ C 2 to B 2 ; C 3 to B$; etc., wherein 
"middle" C is C 3 . This is used by D. C. Miller in the work so 
extensively quoted in this text. 

C_ 2 to B-2I C_i to 5_i ; C to B; c to b; C\ to fa; d to h; c z to fa; 
etc., wherein c\ is middle C. This is used in Germany. The 
French use as C's, t7/, «/, uh> ut 2> uh> etc., where uh is middle C. 

Table VIII 
Musical Intervals 

Interval Name 


Frequency Ratio 




























1 1.000 
81/80 1.013 
25/124 1.042 
16/15 1.067 
27/25 1.080 
10/9 1.1 1 1 

9/8 1.125 

75/64 1.172 

6/5 1.200 

5/4 1.250 

32/25 1.280 

125/96 1.302 

4/3 *-333 
25/18 1,389 
36/25 1.440 

3/2 1.500 
25/16 1.562 

8/5 1.600 

5/3 1.667 
125/72 1.736 

9/5 1.800 

15/8 1.875 

48/25 1.920 

125/64 1.953 

2 2.000 

1 1.000 
1 1.000 
1 1/12 1.059 
1 1/12 1.059 
1 1/12 1.059 
1 2/12 1. 122 
1 2/12 1. 122 
13/12 1.189 
1 3/12 1.189 
1 4/12 1.260 
1 4/12 1.260 
1 S/ 12 *-335 
1 S/ 12 J -335 
1 6/12 1.414 
1 6/12 1. 414 
1 7/12 1.498 
1 8/12 1.587 
1 8/12 1.587 
1 9/12 1.682 
1 10/12 1.782 
1 10/12 1.782 
1 11/12 1.887 
1 11/12 1.887 
2 2.000 
2 2.000 















Semitone or diesis 


Minor second 

Minor tone 

Major second 

Augmented second . 

Minor third 

Major third 

Diminished fourth 

Augumented third 

Perfect fourth ......... 

Augumented fourth .... 

Diminished fifth 

Perfect fifth 

Augumented fifth 

Minor sixth 

Major sixth 

Augumented sixth 

Minor seventh 

Major seventh 

Diminished octave 

Augumented seventh . . . 


14.5. Musical Intervals. — The accompanying table VIII gives 
the musical intervals in a comparative form. This table is taken 
from the preliminary report on acoustic terminology by the stand- 
ardization committee of the Acoustical Society of America. 

It is to be noted that in the last columns is evidence of the 
introduction into the literature of the term " millioctave," which 
is one thousandth of the interval of the octave. 

14.6. Production of Music in the Natural Scale. — Singers and 
players of instruments whose pitch can be regulated by breath or 
touch, find the tempered scale less aesthetically satisfying than 
what they term the "pure" or "natural" scale, and consequently 
endeavor to perform in the latter. Certain choruses in the Ro- 
man and Greek churches do actually use or approximate the 
standards of pitch expounded by Pythagoras, and practice with- 
out instrumental accompaniment in order to form and maintain 
the habit of thinking in terms of this pitch standard. However, 
most performers who talk glibly of the "pure" scale actually in- 
crease major and augmented intervals and diminish minor and 
diminished intervals, as compared with the tempered scale, 
though, in the Pythagorean scale, the major third for instance, 
is not greater, but less, than the tempered interval; while these 
alterations are thus away from, not toward, the natural scale, 
they are aesthetically eminently justified, since in our contem- 
porary music the contrast between major and minor, and the 
"tendency" of certain tones to progress to certain others because 
of our habit of thinking harmonically as well as melodically, are 
of relatively great psychological importance. To distinguish this 
practice of emphasizing "tendency" from a true attempt to ap- 
proximate the scale of nature, I suggest that performers should 
speak of an "artistic" or "harmonic" scale, rather than make an 
inaccurate use of the terms "pure" scale and "natural" scale.* 


1. If the semitone B to C is 1.067, wnv not ta ^ e ^ s as a standard? 
1. What interval is the twelfth root of 1 and why? 
3. If the interval of a whole tone is 1.122, what is the interval of 
a semitone and why? 

* This paragraph has been kindly prepared by Doctor P. G. Clapp, Director 
and Professor of Music at the University of Iowa. To him the author's thanks 
are due. 



15.1. Development of Musical Instruments. — Musical in- 
struments have been developed through experiment. Their use 
has preceded our full understanding of the physics of their opera- 
tion. But with our increased knowledge of acoustics and of psy- 
chology, it is probable that the physicist will have greater influ- 
ence in the development of musical instruments in the future than 
he has had in the past. That improvement can be made both 
by modifications of present instruments and the additions of 
others, there can be no doubt. The purpose of this chapter, in 
harmony with the remainder of the text, is not a detailed descrip- 
tion of musical instruments, nor of the underlying theory. Its 
function is to emphasize physical principles so that the student 
may have his understanding increased and interest aroused. 

At the present time there are excellent treatises * giving dis- 
cussions that are more thorough than are here possible. 

15.2. Production of Sound, General. — The resonance experi- 
ment with a fork placed over ajar, discussed in Chapter V, showed 
that the flow of energy from the fork was increased by the pres- 
ence of the jar. The fork was closely coupled with the air column 
in the jar so that the latter could influence the former. 

This is representative of the method of affecting the flow of 
energy from the source by resonance. Another method is to in- 
crease the vibrating surface exposed to the air. This may be 
illustrated by pressing a vibrating fork on a table top. The table 
becomes a sounding board, and can successfully convey to the air 

* Barton, "Text-book on Sound," Macmillan and Co., 1922, gives both the 
theoretical and practical aspects. Richardson, "Acoustics of Orchestral Instru- 
ments and of the Organ," Oxford University Press, 1929, gives a non-mathematical 
and yet theoretical account. 



more acoustic energy than could the fork directly. These are 
the two general methods used in musical instruments and in sound 
sources in general. 

Returning to the fork placed at the opening of the resonating 
jar in Section 6.5, it is to be remembered that the fork and the air 
column each has its own frequency, and that the intensity of the 
acoustic output, although greatest when these two frequencies 
are alike, yet is noticeably large when the two frequencies are not 
exactly the same. An investigation of the situation shows that 
the resonating air column affects the frequency of the fork rather 
than vice versa. The nature of the change in frequency is curi- 
ous. If the level of the water in the jar is raised, so that the 
natural frequency of the jar is gradually increased in the direction 
of equality to the natural frequency of the fork, the effect is 
opposite to what we would anticipate without resort to mathe- 
matical analysis. The effect is not to change the fork frequency 
in the direction of equality, but in the opposite sense. Then, 
with gradual increase in frequency of the air column, the fork 
frequency continues to shift slowly to higher frequencies. But 
when the natural frequency of the air column becomes equal to 
the natural frequency of the fork, there is a sudden jump in actual 
frequency of the fork back to its natural value with greatly en- 
hanced intensity of resonance. The entire variation of frequency 
of the fork in a certain experiment performed as described was 
less than one-hundredth of one per cent. The effect is therefore 
not large. The experiment is cited to show that a vibrating air 
column can affect the frequency of such a rigid body as a tuning 
fork. This effect of one of the coupled vibrating systems upon 
the other, and the increase of flow of energy from the source at 
like frequencies, are the two most prominent effects to bear in 
mind. There are cases when the source is coupled to more than 
one vibrating system. This will be described at the appropriate 

There is another feature that should be mentioned in stating 
a general view. One is concerned not only with the intensity of 
sound on the interior of an instrument, but with that on the ex- 


terior. In wind instruments with open orifices the transfer is 
very complicated. 

15.3. Production of Sound by Strings. — Stringed instruments 
are actuated by impact, plucking and bowing. In the first two 
cases, the string is forced into a strained position and then allowed 
to vibrate freely. Since the string is capable of vibrating in any 
of its natural frequencies, which are integral multiples of the 
fundamental one, the relative amplitudes which the various com- 
ponents have will depend upon the original displaced position of 
the string. For example, assume that by a carefully constructed 
constraint it is easily possible to displace and hold the string 
throughout its entire length in that position corresponding to a 
displacement for the fundamental frequency alone. From this 
arbitrary position the string is suddenly released. Clearly it 
would then vibrate freely with only the fundamental frequency 
involved. Similarly, the string might be displaced to a position 
corresponding to the actual position of the string when possessing 
the frequencies of the fundamental and the first overtone. Then 
the subsequent vibrations would contain only these two frequen- 

When a string is plucked or is struck by a hammer, the dis- 
placement produced is much more complicated, and, moreover, 
the two cases would not be alike. In the case of plucking, the 
entire string is set free from practically a rest position. On the 
other hand, a blow of the hammer is very rapid and is actually 
completed before the string has had time everywhere to move into 
a displaced position. These two different starting conditions of 
the string will result in a difference of the values of the amplitudes 
of the component tones in each case. 

For a similar reason, one can understand why the size and 
softness of a hammer would cause a difference in the quality of 
tone from a string. The hard hammer would give a sharper bend 
to the string. One associates sharp bends with short distances 
between nodes and hence with higher frequencies, and a careful 
study shows that such is really the case. The hard hammer 
accentuates high frequencies. 


From the above discussion, it is rather clear that the quality 
of tone can be modified by the speed of plucking (which may not 
be a practical consideration), the nature of the hammer, the 
quickness of its blow, and the position along the string of either 
the plucking or the impact. But it happens that in the hammer 
instrument, the piano, the actuating key does not determine the 
quickness of the blow. After the hammer is in motion, the key 
no longer has any control. Give the hammer a certain energy or 
a certain velocity and it will always strike in the same manner. 
Thus the possibilities of "touch" on the piano are seriously lim- 
ited, much more so than is commonly believed. 

The quality of a bowed string can be modified at will much 
more readily. Bowing presents a complicated and an interesting 
phenomenon. The bow pulls the string to one side, the force 
arising from what is called "static friction." It is well known 
that such a force is greater than that existing when slippage once 
begins. So when the bow pulls the string to one side, the string 
finally slips under the bow. When the slip once begins it con- 
tinues until the string has reached a displacement in the opposite 
direction. Then the string again follows the bow's motion and 
repeats its former action. Work is done by the bow on the string 
because the force acting in the direction of the bowing is very 
much greater in one half of the vibration than in the other half. 
From this description it is evident that the width of the bow, the 
place of bowing along the string, the pressure on the bow and the 
speed of bowing will all enter into the production of the tone 
quality. The possibility of "touch" in such an instrument is 
therefore relatively large. 

15.4. Production of Sound by Reeds. — In the clarinet there 
is but one reed, and in the oboe and bassoon there are two. The 
reeds are set into motion by blowing. The velocity of the air 
blast lowers the pressure between the reed and its base, or be- 
tween the two reeds, and the impact of the air behind also con- 
spires to cause the closing of the reed or reeds. The pressure not 
being sufficient to keep the reed or reeds in a closed position, the 


original position is resumed and a continuous vibration ensues. 
In this vibration the reeds, acting alone, have a natural frequency, 
or rather a group of natural frequencies, with a quality of sound 
which is not very pleasing. But the reeds are usually, as in the 
clarinet and oboe, coupled with an air column. With such an air 
column and a thin reed, the former is the chief factor in deter- 
mining the frequency of the coupled system. It is to be borne in 
mind that reeds are used in organ pipes as well as in orchestral 

But there is another factor, the importance of which is not 
definitely known. The resonating cavity in the mouth of the 
blower is also coupled to the vibrator. It is taught by some 
authorities that the blowing of an instrument is made easier if 
one will put his mouth in the position of humming the note de- 
sired. Until quantitative data are obtained, a positive statement 
concerning the importance of this second coupling cannot be 


What has been stated about the reeds would also apply to the 
use of the lips with brass instruments, except in the latter case 
there is a large opportunity for the alteration of the vibrator. 
The lips may be stretched more or less tightly and they may be 
thrust forward. The shape of the mouthpiece in the case of 
brass instruments and its influence on the quality deserves addi- 
tional mention in a later section. 

15.5. Production of Sound by an Air Blast. — The most com- 
mon examples of the production of sound by an air blast are the 
blowing of the flute and of an organ pipe. If a sheet of air strikes 
a sharp edge a tone is produced which depends upon the shape of 
the edge, the velocity of the blast, and the distance from the blast 
opening to the edge. This tone is caused by the production of 
eddies first on one side of the sharp edge and then on the other. 
When such a source is coupled with a resonating air column, the 
frequency of the eddies or vortices is controlled thereby. The 
details are beyond the scope of this text, but enough has been 
said to remind the reader that here again one has two vibrating 


systems, with one controlling the other. The natural frequency 
of this "edge tone" is raised by an increase in air velocity and is 
lowered by an increase in distance of the edge from the air blast 
opening. The production of the edge tone can therefore be con- 
trolled, by the velocity of the air blast and the distance of the 
blast opening to the edge against which the blast impinges. This 
is shown in practice in the use of the flute and in the manufacture 
of organ pipes. 

There is one applic ation of the air-blast in which its importance 
is not quantitatively known. In most brass wind instruments 
the mouthpiece is cupped so that there is an edge at the opening 
of the tube of the instrument into the cup. The blast of air from 
the lips must impinge upon this edge. There is here a secondary 
source of sound produced. This may be the explanation, at least 
in part, of the effect of the shape of the mouthpiece upon the 
tone produced in such an instrument. 

15.6. Harmonics and Overtones. — In dealing with the pro- 
duction of sound, one must have clearly in mind the possibility 
of regarding every sustained sound as composed of various fre- 
quencies, each of which is strictly a simple harmonic vibration. 
Any vibration which repeats itself can be analyzed into its component 
simple harmonic frequencies. In this analysis the frequencies of 
the components may include those whose relative frequencies are 
1, 2, 3, 4, 5, 6, etc. But these frequencies may not all be present. 
The amplitudes of some of them may be zero. If the one repre- 
sented above as "1" is present, all of the others present are in 
consonance with it, for each represents an integral number of 
octaves. Such overtones are called "harmonics." But if the 
fundamental, or lowest tone present, is represented by 2, then, 
while 4, 6, 8, etc., which are octaves of the first, may be called 
harmonics, yet 3, 5, 7, etc., are not octaves and to them is given 
the more general term "overtones." Thus the overtones may 
in this narrow sense be harmonic and they may not be. Also the 
overtones may be dissonant with the fundamental. For assume 
the lowest tone present is that represented by 8. Then overtone 


9 would be dissonant with it. Every musical sound, then, has 
a fundamental, or the lowest tone, but the overtones need not be 
harmonics. This is true, for example, of most wind brass instru- 
ments. In them the tone represented in the above by " 1 " is not 

15.7. Peculiarity of Action of Several Instruments. — In the 

violin, the vibration of the string is practically parallel to the body 
since the displacements are along the direction of the bow. This 
vibration causes a motion of the bridge in the same direction. 
Under the bridge, and more nearly under one foot of the bridge 
than the other, there is a supporting sound-post. About this side 
which is rather stable, the bridge rolls back and forth with the 
vibration of the string. There is thus conveyed to the body of 
the violin, and indeed to both the belly and back of the instru- 
ment, a vibration perpendicular to both. The sound from the 
instrument thus depends upon the natural frequencies of the 
violin itself and the air volume within. But the radiation of 
energy from the instrument is largely enhanced by the sounding- 
board effect, or the increase of exposure to the air of the vibrating 
surface. Thus the vibrating string is coupled with resonating 
bodies and also with an increased area which is forced to vibrate. 
That the natural frequencies of the violin are of importance is 
shown by the difference in instruments. H. Backhaus * has ex- 
perimented with famous violins and has shown that one of their 
chief characteristics is emphasis upon the high overtones, or upon 
the actual number of overtones that have a measurable ampli- 
tude. A great deal may be said about the tone characteristics 
of violins and their construction, but this would take the reader 
too far afield. Attention may be called, however, to the fact that 
Norway spruce, which is frequently used in violin construction, 
has a great elasticity and small mass per unit volume. This gives 
a high velocity (15,000 feet per second) of sound and causes the 
vibration to spread rapidly over the violin. Of course the veloc- 
ity across the grain is, unfortunately, much less. 
* Die Naturwissenschaften, 18, 1929. 


In the piano we find one bridge on the sounding board and one 
on the frame which carries the tension in the wires. The vibra- 
tions are conveyed to the sounding board by the former. The 
hammers are made to strike at one-seventh the length from one 
end. This location would help to prevent the formation of the 
seventh harmonic which is somewhat dissonant, but it is thought 
by Richardson (loc. cit.) that this is not the most important effect. 
In his opinion, if the string is struck at one-seventh from the end, 
its fundamental amplitude has the maximum value and conse- 
quently less of the energy can go into the high harmonics. Hence 
the excitation of natural vibrations of the sounding board are re- 
duced to a minimum. 

The clarinet is a cylindrical tube terminating in a bell. It is 
usually stated that the fundamental and harmonics have the rela- 
tive frequencies of I, 3, 5, 7, etc., as is the case with a cylinder 
closed at one end, for the mouthpiece acts as a closed end, though 
it is the source of sound. But it is possible that the mouthpiece 
may be made to act as an open end introducing noticeable even 
harmonics as well. There is a vent hole in the instrument near 
the beak which is opened in order to play the higher notes in its 
range. The function of this small hole is not very clear. It is 
claimed that it helps in the formation of an antinode near the 
beak end, and thus encourages the formation of high tones. But 
such an orifice would also be favorable to the transmission of the 
higher frequencies, as pointed out in Chapter XIII, and would 
thus have the tendency actually found in the use of the instrument. 

The oboe has a conical tube, with a "closed" end at the reeds. 
Its natural frequencies include therefore all of the harmonic series. 

Many brass instruments have a hyperbolical or an exponential 
shape. In these the tube widens out slowly at first and then very 
rapidly at the bell. The harmonics are the same as those of a 
conical tube. The bell on these instruments, as on the clarinet 
and oboe, enables the energy from the interior to escape more 
rapidly than otherwise. But the energy may escape from the 
opened holes also, as described in the following Section 15.8. 

The organ pipe must be considered as open at the air-blast 


end. Thus the harmonics and fundamental have the variation 
of 1, 3, 5, 7, etc., for a pipe closed at the other end, and all the 
harmonics for an open end. In measuring the acoustic length, 
or the length referred to in Section 6.8, of an organ flue pipe, a 
correction must be made for the open end, and also for the mouth 
or air-blast end. The former correction has been mentioned in 
Section 6.9. The latter has recently been studied by Bate.* He 
found that the correction was proportional to the area of the pipe, 
inversely proportional to the area of the mouth, and independent 
of the frequency over the octave studied. 

15.8. Emission of Sound from the Clarinet. — It is the purpose 
of this section to emphasize the very complicated action within 
an instrument such as a clarinet and the effect upon its emitted 
sound. A later section discusses the frequencies produced by 
opening the holes. But up to the present time no one has studied 
in detail the emission of sound from the instrument. Not only 
does the opening of the holes modify the resonance inside, but a 
pipe having open holes will inevitably have a filtering action in- 
side, as discussed in the previous chapter. The sound in part 
issues from the opened holes as well as from the bell and with 
some notes the former is of high importance. In the absence of 
experimental facts, one can surmise that the quality of these two 
sounds would not be the same. The entire action, as one readily 
sees, is very complicated, for one has to deal at least with reso- 
nance, the filtering action of one or more orifices, waves travelling 
in both directions along the tube, radiation from the orifices and 
the affect of return waves upon the source. 

15.9. Production of the Voice. — The vocal cords or ligaments 
require a blast of air from the lungs to pass through the slit be- 
tween them-t They act in a manner similar to two non-rigid 
reeds. The frequencies are determined by tension, length and 
distribution of mass of the cords. The resonances of the larynx, 

* Bate, Philosophical Magazine, 10, 65, p. 617, 1930. 

t R. L. Wegel, Jl. Acous. Soc. of Am. y 1, 3, p. 1, I93°> discusses the "Theory of 
1 the Vibration of the Larynx." This paper is of interest particularly to physicists. 


pharynx, mouth and nose control the harmonics that are empha- 
sized and give the quality to the voice. As earlier stated, the 
differences in sustained vowels are caused by resonance. But it is 
also evident that resonance cannot fully control quality; for sus- 
tained tones, only the vibrations present in the vocal cords can 
exist in the tone produced. For the impulsive sounds which also 
proceed from the vocal cords, the quality is produced by the exci- 
tation of all the natural frequencies of the resonating chambers. 
This action is similar to the effect of sending a puff of air across 
the opening of a bottle. The natural frequencies die out rapidly. 
So, in speech and music both types of action are prominent. 

15.10. Frequency of Pipes. — It has been well known that the 
frequency of a pipe, a flute for example, with one hole open, de- 
pends upon the resonating action of the column of air extending 
from the opened hole to the source of sound. In the case of the 
flute the source is the hole across which the player blows. If the 
frequencies are computed by assuming that the effect of the 
opened hole is to create a pipe with an open end at that point, 
then, since the blown end is also effectively an open end, the 
wave-length is twice the distance from the source to the open 
hole. This is only approximately true, and for the reason that 
the opened hole is not equivalent to an open end. 

If two holes on the instrument are open, then the resonance 
frequency is modified. An examination of the theory * shows 
that now the frequency can be calculated by assuming the pipe 
to consist of two resonant systems consisting of a pipe from the 
source to the first hole, and a pipe from the first hole to the second. 
In fact, it is possible by making approximations in the theory to 
compute the frequency for the case of any number of open holes. 
Yet these approximations do not permit of manufacturing designs 
based solely on computations. Even now the location of holes 
depends upon experiment. 

z5.11. Aeolian Harp. — When wind blows over stretched wires 
sounds are produced which are not due to the vibrations of the 

* Irons, Phil. Mag., 10, p. 16 (1930). 


wires but to the action of the air itself. When the air passes an 
obstacle, vortices or whirls may be established on the leeward 
side. It is the instability of these vortices that leads to oscilla- 
tions and finally to sound production. The predominant tone 
has been found by recent observations* to have the following 

n = 0.20^, 

wherein v is the velocity of the wind and d the diameter of the 


Of course if a natural frequency of the string corresponds to 
that given in the above formula, the tone is reinforced. 

The noise of the wind at corners, in the trees, at all obstacles 
is caused by the instability of vortices and consequently the pre- 
dominant term depends upon the shape of the obstacle and the 
velocity of the wind. The "howling" of the wind is the char- 
acteristic sliding of the tone in pitch which is precisely what would 
be caused by continuously varying wind velocities. 

15.12. Singing Flames. — If a small flame is introduced into 
an open pipe, it may excite the natural vibration of the pipe. 
Its position must be between the open end and a node, or about 
half way to the mid-point of the pipe. When the frequency of 
the pipe is excited the height of the flame oscillates simultane- 
ously with changing pressure, the pressure in the gas and in the 
air being in opposite phases. Heat is transferred to the air at 
each condensation.! 

15.13. Singing Tubes. — Recently Professor C. T. Knipp has 
made an apparatus which exhibits the phenomenon in a conveni- 
ent manner. A heated glass bulb is the source of energy and the 
tone is the natural vibration of the air in the chamber volume 
to which it is attached. One can tune this volume and thus secure 
a fairly constant source of sound. 

* Richardson, Physical Society Proceedings, 36, 153, 1924, and 37, 178, 1925. 
f See Barton, he. tit., article 265, et seq. 


15.14. Sensitive Flames and Jets. — When a fluid jet issues 
from an orifice into quiet air, the pressure behind the jet may 
become so excessive as to cause an unsteady state in the jet. If, 
however, this pressure is not quite attained, a sound wave of 
certain frequencies may produce this unsteadiness. In the case 
of a flame, a flare results. A sensitive flame responds to sound 
waves because of the displacements of the air and not because of 
pressure changes. The action occurs near the orifice. 

15.15. Tones from Membranes. — In acoustical apparatus 
membranes and diaphragms are in common use. Some of the best 
telephone transmitters for wireless broadcasting use a stretched 
membrane (or very thin flexible plate) and a vibrating plate (not 
stretched, but whose stiffness is caused by its thickness) is uni- 
versally used in telephone receivers. To illustrate the nature of 
the natural vibrations of such pieces of apparatus, a brief descrip- 
tion will be given of the modes of vibration of an ideal circular 
membrane. The assumptions are that the membrane is stretched 
uniformly in all directions with a certain tension that remains 
unaltered during the vibrations considered, and that the mem- 
brane is perfectly flexible and infinitely thin. A membrane is 
"perfectly flexible" if there is no force resisting bf nding. The 
membrane, under the assumptions, is fastened rigidly at its cir- 
cular boundary. Its fundamental vibration is with the circum- 
ference as a node and a maximum displacement at the center of 
the circle. Inasmuch as our interest is in the ratios of the funda- 
mental and overtones, it is satisfactory to represent all frequen- 
cies as multiples of the fundamental. Thus the frequency of the 
fundamental is regarded as unity. In Fig. 15.1 * are shown the 
nodal lines and the frequencies of the fundamental and eleven 
overtones. The first number under each figure is the frequency. 
The second and third numbers are the radii of the nodal circles 
as compared with the radius of the membrane. In each figure 
the nodal lines are those belonging to that overtone only. The 
chief points of interest are first, that the overtones are not mul- 

* This figure is according to Rayleigh, " Sound," p. 206. 



tiples of the fundamental and second, that the actual vibration 
of a membrane, when involving all these tones, is very complex. 
The vibrations of plates are not the same as those of mem- 
branes because the stiffness in the former has a different origin; 

Fig. 1 5. 1 

namely, the rigidity of the plate and not the tension in it. Yet 
the general description of the nodal system as consisting of con- 
centric circles and symmetrically distributed diameters remains 
correct. In the ordinary telephone transmitter diaphragm the 
fundamental has a frequency in the neighborhood of 800 cycles. 
A force having this frequency would set the diaphragm into a 
relatively large vibration. Reference to the discussion of the dis- 


tribution of energy in speech will show that the maximum energy 
does not occur at 800. Experience has shown that this frequency 
gives the most satisfactory results when all the practical consid- 
erations of design are met. 

Reference should be made to the sound of a drum. Its pitch 
is difficult to get, except with the kettle drum, where the pitch 
is modified at will by altering the tension. The quality of the 
sound of the drum is caused by the fact that its overtones are 
somewhat dissonant with each other. 

15.16. Sound Waves in a Solid. — If a small rod is stretched or 
compressed longitudinally, there is an accompanying contraction 
or expansion laterally. There is therefore a change of shape of 
the material as well as a longitudinal dilation or compression. 
But if the solid is extended in all directions, this change of shape 
laterally is not free to occur and the velocity of the sound wave 
is not the same. The velocity of a longitudinal wave in an ex- 
tended solid can be expressed by an equation similar to (1.2) of 
Chapter I, but E is then not a single constant but the sum of two. 
One of these elastic constants deals with volume elasticity and 
the other with shape elasticity. 

But if instead of using a longitudinal force at the end of a rod, 
causing longitudinal waves, we had used a force twisting the rod 
first one way and then the other, we would have caused a wave 
of twist or a torsional wave in the rod. Thus it is possible to 
have in an extended solid a second kind of wave. Its velocity 
is expressed by an equation similar to that used for the longitu- 
dinal wave, E, depending upon the same two constants as before, 
but not in the same manner. The velocities of the two waves are 
not equal. Hence if we had spherical waves of both types start- 
ing from a point, one wave would travel faster than the other. 
In fact, the elastic constants are such that in most materials the 
velocity of the longitudinal wave is roughly twice that of the tor- 
sional wave. 

Then it is possible also to produce transverse waves in a solid. 
Rest a thin horizontal bar on two edges placed approximately 



0.224 of the length from each end. It will vibrate transversely 
with these edges as nodes, and the ends of the bar as loops. It 
will give a relatively pure tone because the natural frequencies 
of the rod are widely separated and, moreover, the fixed position 
of the nodes makes possible only a few of these. In fact, experi- 
ment and theory both show that the first overtone that would 
have a node at the same point has 13.3 times the fundamental 
frequency, or between three and four octaves above the funda- 
mental. The position of this node is approximately at 0.226. 

A tuning fork is a rod bent into a U shape, but with the ad- 
dition of a mass of metal at the bend. Also the nodes of the 
bar approach each other as the rod is bent. The first overtone 
of a bar bent into a U shape is about six times the fundamental 
frequency. The wide separation of the fundamental from the 
first overtone and the relative faintness of latter are the reasons 
for the purity of the tone of the tuning fork. 

15.17. Vibration of Bells. — A bell may be regarded as a vi- 
brating cylinder. Consider Fig. 15.2 showing an exaggerated 
cross-section perpendicular to the axis of the cylinder. Assume 
that the bell experiences its fundamental vibration from one 
approximate ellipse to another 
with perpendicular axes. If the 
reader will draw the circle and 
the two ellipses he will see that 
the four "nodal" points on the 
circumference are not stationary. 
The points of intersection of one 
ellipse and circle are not the 
same as the crossing points of 
the other ellipse and circle. If 

there were stationary nodes we might assume that transverse (or 
torsional) waves were the only ones existing in the bell. But, 
under the circumstances, both longitudinal and transverse waves 
exist. It can be shown, as may be here surmised, that the over- 
tones are not harmonics. (See Section 15.18.) 

Fig. 15.2 


The physical requisites of a good bell are that the material 
be homogeneous, relatively free from viscosity and have an elastic 
limit * that will not be exceeded in ordinary use. The funda- 
mental tone of the bell is not prominent. In fact, it does not 
determine the pitch of the bell. The pitch of the strike note is 
an octave below the fifth partial or the fourth overtone, and in a 
bell not intrinsically tuned (see Section 15.18), does not corre- 
spond to any actual vibration of the bell. The cause of this judg- 
ment as to the pitch of a bell is not understood-! 

15.18. Carillons and Chimes. — There seems to be some loose- 
ness of definition in the literature concerning the terms "carillon" 
and "chime." Both refer to a set of bells tuned to a musical 
scale, but in the carillon all the semitones appear. It is obvious 
that such bells must be carefully tuned to pitch. This is done 
by casting each bell for a slightly higher tone than desired and 
then by removing some of the casting by cutting and abrasion 
with the aid of machinery. Bell founders have for many years 
considered the seven desirable tones of a bell to have the following 
frequency ratios, 1 : 2 : 2.4 : 3 : 4 : 5 : 6. But this result cannot 
be secured except by the most skillful application of the art. In- 
deed, at the present time, a bell ranks with the best if only the 
fundamental and the first four overtones with the ratios stated 
are obtained. Even this requires machining in a manner that is 
regarded as a trade secret. 

But bells with tuned overtones are unusual and occur only 
in the most expensive carillons. An illustration of what is con- 
sidered a good bell will now be cited. Dr. A. T. Jones has care- 
fully measured the frequencies of the largest bell of the Dorothea 
Carlile Chime at Smith College J and has found that the first 

* An elastic body when stretched will return to its original position when the 
stretching force is removed, provided a certain limiting tension is not exceeded. 
This limiting tension per unit area is called the elastic limit. 

t See article by A. T. Jones, Physical Review, XVI, 4, 1920, p. 247, and Journal 
Acoustical Society of America, I, p. 373, 1930. 

% A. T. Jones, Jl. Acous. Soc. Am., I, p. 373, 1930. 


seven partials * have the ratios 1.00 : 1.65 : 2.10 : 3.00 : 3.54 
: 4.97 : 5.33, which are very different from the ratios given above 
for a bell with tuned overtones. It is surprising that a bell like 
the one described could be pleasing to the ear. But it must be 
remembered that harmonics of the seven tones are not present, 
and consequently the dissonance that would occur in sounding 
the same tones on a string instrument is not observed. The bells 
with tuned overtones are, however, a distinct improvement and 
will be increasingly used. 

15.19. Acoustic Power Output. — Dr. P. E. Sabine has meas- 
ured the acoustical power output of certain musical instruments. 
A violincello in its fundamental, when bowed strongly, varied from 
100 microwatts (or 1,000 ergs per second) at 128 cycles to one 
microwatt at 650 cycles. A good violin gave a fairly uniform 
output of 60 microwatts for frequencies from 192 cycles to 1,300 
cycles. Open diapason organ pipes gave an output of approxi- 
mately 1,000 microwatts. All the values refer to the fundamental 
tone and not to the entire output. 

15.20. Modern Loud Speakers. — This text has shown that 
with closely coupled systems one may get much more energy from 
the source of vibration with than without the condition of reso- 
nance. Moreover, it has been recognized that any mechanism 
having elasticity and inertia may be likened to a spring and a 
weight suspended therefrom. There is a natural period of vibra- 
tion. Modern loud speakers frequently have a cone-shaped sur- 
face, actuated electrically at its vertex by the complex vibrations 
which eventually are to be given to the air. The vibration at 
the vertex cannot convey sufficient sound directly to the air. 
The cone is employed because of its large exposure to the air. 
It is a sounding board. But being light and far from rigid, the 
cone has natural frequencies of its own. Thus in actual use, any 
of the cone's natural frequencies occurring in the original sound 
at the transmitting station will be overemphasized by the loud 

The term "partial" is used to include both the fundamental (first partial) 
and the overtones. 


speaker. Obviously the cone could be made of very rigid mate- 
rial, so rigid that its natural frequencies would be higher than any 
used at the transmitting station. But then it would be too mas- 
sive to actuate. One of the difficulties is to obtain a cone light 
enough, large enough and yet without introducing a distortion of 
the complex sound. Another important feature of interest is the 
use of a plane, rim, or surface surrounding the base of the cone. 
It may be called a baffle plate. It has three functions. It sepa- 
rates the inside of the cone from the outside, thus preventing the 
slippage of air from the outside of the cone to the inside. Such 
a slippage could decrease the pressure caused by the vibration 
of the cone and hence the intensity of the sound wave. The 
baffle plate also reflects the sound. In Section 5.5 it is shown 
that by this reflection a greater amount of sound issues from the 
vibrator. The third function is that of shutting out the vibration 
from the other side of the cone. This is desirable for one can 
readily see that the vibrations on the two sides are opposite in 
pressure phase and will produce interference. 

Horns, usually with diameters increasing much more rapidly 
than the length, are used in loud speakers in theaters and public 
address systems. They are coupled systems in which the horn 
and the moving diaphragm each has its own natural frequencies. 
The endeavor is made to reduce these resonance effects to a mini- 
mum. A practical discussion of horn design is given by Hanna.* 
He favors a horn of the "exponential" type, for theory points 
approximately to this shape. An exponential horn doubles its 
area at equal intervals along its length. 


1. What mechanical conditions must be met if a musical instru- 
ment is to give great intensity of sound? 

2. In what ways is the transfer of energy from the initial vibrating 
portion of a musical instrument to the atmosphere accomplished? 

3. Under what conditions are two vibrating systems "coupled?" 
Give an illustration. 

4. Give an illustration where three vibrating systems are coupled. 

* Hanna, Journal Acoustical Society of America, II, 2, p. 150, 1930. 


5. By what means (ideal and not practical) can a string be set 
into vibration of the fundamental alone? 

6. In what way does the nature of the piano hammer affect the 
quality of tone? 

7. While one holds a piano key down, what is the position of the 
hammer, of the damper? 

8. Discuss the possibilities of the qualities of a tone in a brass 
instrument and the ease of blowing being altered by the adjustment 
of the mouth. 

9. How do you know that the vibration communicated to the 
body of a violin is not one caused by the changes in tension of the 
string during its vibration? 

10. What is the reason for using, in the violin, a wood having a 
high sound velocity for transverse displacements? 

11. Physically how can tone quality be modified? 

12. Upon what does the quality of the voice depend? 

13. State the difference in action of the sustained and the impul- 
sive sounds of the voice. 

14. Explain the howling of the wind. 


Absorption, 26-33 
Absorption coefficients, 30, 32 
Absorption along a conduit, 78 
Aeolian harp, 186 
Air blast production, 181 
Airplane sounds, $6 
Architectural acoustics, 151 
Asymmetry of vibration, 135 
Asymmetrical vibration of ear, 


Audibility, 123-139 
Audibility, limits of, 124 
Auditorium acoustics, 27-34, 

Auditor, diffraction about, 61 

Baffle plate, 193 
Beam of sound, 37 
Beats, 36 } 148 
Bells, vibrations of, 191 
Binaural beats, 148 
Binaural effects, 140-148 
Binaural intensity effect, 140 
Binaural phase effect, 142, 146 
Blend of sounds, 139 
Branch tubes, 164 
Buildings, resonance in, 95 

Carillons, 192 
Characteristic vowel 

Chimes, 192 


Clarinet, 184, 185 
Clearness of enunciation, 114 
Coefficient of absorption, 31 
Combination tones, 135, 137 
Complex tones and binaural 

effect, 146 
Conduits, 153, 155, 157 
Conical megaphones, 92 
Constriction in conduits, 153 
Coupling, 177, 178, 181 

Deafness, 125 

Decay of intensity in pipes, 156 
Decibel, 129 
Density, 16 
Diatonic scale, 172 
Difference of phase at ears, 142 
Diffraction, 57-63 
Doppler's principle, 20 

Echo, 25 

Emission of sound and reso- 
nance, 87 
End correction of pipes, 91 
Energy of a wave, 26, 68, 74 
Energy distribution in speech, 

117, 119 
Expansion in conduits, 153 

Fechner's law, 129 
Filters, 168 
Flames, singing, 187 




Flute, 186 

Frequency, 5, 20, 174, 186 

Fundamental, overtones, 99 

Harmonic motion, simple, 12 
Harmonics, 182 
Hearing, noise present, 132 
Helix, wave in, 7 
Helmholtz resonator, 88, 165 
Herschel Quincke tube, 159 
Horns, 47, 92 
Huyghens' principle, 36 

Image, 25, 70, 71 
Instruments, musical, 95, 177- 

Instruments, sounds from, 101 
Intensities of fundamental and 

overtones, 99 
Intensity, 26, 123, 127, 128, 

129, 130 
Intensity of tone-blend, 139 
Interference, 35, 40, 159 
Intermittent tones, 138 
Interval, 173, 176 

Least time, principle of, 63 
Limits of audibility, 124 
Localization of sound, 146 
Longitudinal vibrations, 18 
Loops, 83, 85 
Loudness, 127 

Loudness of complex sounds, 135 
Loudspeaker, 193 

Machinery noises, 151 
Masking effect, 130 

Mean temperament, 173 

Megaphones, 92, 93 

Membranes, 188 

Minimum audibility, 123 

Minimum perceptible fre- 
quency difference, 133 

Minimum perceptible intensity 
difference, 130 

Minimum time for tone per- 
ception, 133 

Mirror, parabolic, 39 

Musical instruments, 177-194 

Musical instruments, resonance 
in, 95 

Musical tones, 97 

Natural scale, 176 

Nodes, 83, 85, 90, 91, 94, 170 

Noises of machinery, 151 

Oboe, 184 
Organ pipe, 137, 185 
Orifice, 166 
Overtones, 99, 182 

Parabolic mirror, 39 
Paracusis, 132 
Particle velocity, 21 
Perception of tone, 133 
Phase, 35 
Phase change, 65 
Phase difference, 44 
Phase difference of compo- 
nents, 104 
Piano, 184 

Pinnae, reflection from, 46 
Pipes, 84, 90, 91 



Pitch of sound-blend, 139 
Power output, 193 
Pressure, 11 

Pressure on reflector, 138 
Production of sound, 1 77-1 81 

Quality, change by diffraction, 

Quality, instrumental, 101 

Reciprocal theorem, 61 
Reflection, 23, 65-78 
Reflection at a change in area, 

Reflection at a closed end, 77 

Reflection at an open end, 75 

Reflection in gases, 68 

Reflection of ripple waves, 6 

Reflection, total, 52, 77 

Reflection with change of 
phase, 66 

Reflection without change of 
phase, 65 

Reflector, plane, 38 

Reflectors, 38-47 

Refraction, 50-57 

Resonance, 80-96 

Resonator, Helmholtz, 88 

Resonance of the voice, 90 

Resonance in cylindrical pipes, 

Resonance in conical pipes, 92 

Resonance in musical instru- 
ments, 95 

Resonance in buildings, 95 
Reverberation, 26-31 

Scale, diatonic, 172 
Scattering of airplane noises, 56 
Selective reflection, 42 
Semitone, 173 
Sensation level, 129 
Sensation units, 128 
Sensitive flames, 188 
Sensitivity of the ear, 123 
Silence areas, ^ 
Singing flames, 187 
Singing tubes, 187 
Solid, waves in, 16, 190 
Speech energy, 121 
Speech, physical factors of, 

1 17-122 
Speech sounds, 106 
Stationary waves, 81-83 
Stationary waves in a closed 

pipe, 84 
Stationary waves in general, 94 
Stethoscope, 154 
String, vibration of, 97, 179 
Stringed instruments, 179 

Temperament, 173 
Temperature, effect of, 19, 49 
Tone, whole, 173 
Transmission, 149-158 
Transmission in buildings, 151 
Transmission, selective, 159- 

Tubes, singing, 187 

Velocity of a wave, 9, 15, 19, 

Velocity of a particle, 21 

2oo INDEX 

Velocity in pipes, 156 Waves, 3 

Vibrato, 134 Waves properties of, 4 

Violin, 183 Wave niters, 168 

Viscosity, 26 Wave length, 20 

Voice, production, 185 Wave, sound, 10 

Voice, resonance of, 90 Weber's Law, 128 

Vowel sounds, 106-116 Wind, effect of, 52, 53 



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